It is a key result that the composite $$ MU_* \xrightarrow{h} H_*(MU;\mathbb Z) \xleftarrow[\sim]{\Phi^{\vee}}H_*(BU;\mathbb Z),$$

where $\Phi^{\vee}$ is the dual of the Thom isomorphism $\Phi$, agrees with evaluating on normal Chern numbers.

In other words, $\langle \Phi(c), h([M])\rangle = \bar c(M)$ for all $c \in H^*(BU)$ and for all $[M] \in MU_*$.

(A reference in the real case is the diagram on page 228 of A concise course in algebraic topology by J.P.May. The complex case is identical.)

So the assertion is just a Chern number calculation:

$h([\mathbb CP^n]) = (n+1)m_n$ if and only if, for all $c \in H^{2n}(BU;\mathbb Z)$, $$\langle \Phi(c), (n+1)m_n\rangle = \bar c(\mathbb CP^n).$$

It is a key result that the composite $$ MU_* \xrightarrow{h} H_*(MU;\mathbb Z) \xrightarrow[\sim]{\Phi^{\vee}}H_*(BU;\mathbb Z),$$

where $\Phi^{\vee}$ is the dual of the Thom isomorphism $\Phi$, agrees with evaluating on normal Chern numbers.

In other words, $\langle \Phi(c), h([M])\rangle = \bar c(M)$ for all $c \in H^*(BU)$ and for all $[M] \in MU_*$.

(A reference in the real case is the diagram on page 228 of A concise course in algebraic topology by J.P.May. The complex case is identical.)

So the assertion is just a Chern number calculation:

$h([\mathbb CP^n]) = (n+1)m_n$ if and only if, for all $c \in H^{2n}(BU;\mathbb Z)$, $$\langle \Phi(c), (n+1)m_n\rangle = \bar c(\mathbb CP^n).$$

It is a key result that the composite $$ MU_* \xrightarrow{h} H_*(MU;\mathbb Z) \xleftarrow[\sim]{\Phi}H_*(BU;\mathbb Z),$$

where $\Phi$ is the dual of the Thom isomorphism, agrees with evaluating on normal Chern numbers.

In other words, $\langle \Phi(c), h([M])\rangle = \bar c(M)$ for all $c \in H^*(BU)$ and for all $[M] \in MU_*$.

(A reference in the real case is the diagram on page 228 of A concise course in algebraic topology by J.P.May. The complex case is identical.)

So the assertion is just a Chern number calculation:

$h([\mathbb CP^n]) = (n+1)m_n$ if and only if, for all $c \in H^{2n}(BU;\mathbb Z)$, $\langle \Phi(c), (n+1)m_n\rangle = \bar c(\mathbb CP^n)$.

It is a key result that the composite $$ MU_* \xrightarrow{h} H_*(MU;\mathbb Z) \xleftarrow[\sim]{\Phi^{\vee}}H_*(BU;\mathbb Z),$$

where $\Phi^{\vee}$ is the dual of the Thom isomorphism $\Phi$, agrees with evaluating on normal Chern numbers.

In other words, $\langle \Phi(c), h([M])\rangle = \bar c(M)$ for all $c \in H^*(BU)$ and for all $[M] \in MU_*$.

(A reference in the real case is the diagram on page 228 of A concise course in algebraic topology by J.P.May. The complex case is identical.)

So the assertion is just a Chern number calculation:

$h([\mathbb CP^n]) = (n+1)m_n$ if and only if, for all $c \in H^{2n}(BU;\mathbb Z)$, $$\langle \Phi(c), (n+1)m_n\rangle = \bar c(\mathbb CP^n).$$