Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible operators in $B(H)/K(H)$, $Inv(B(H)/K(H))_0$ --- connected component of $id$ in $Inv(B(H)/K(H))$. $ind\colon Inv(B(H)/K(H))\to \mathbb{Z}$ --- Fredholm index.

I want to find a reference for the following fact:

Fact 1. If $H$ is infinite-dimensional and separable then $ind$ is locally-constant and provides an isomorphism between $Inv(B(H)/K(H))/Inv(B(H)/K(H))_0$ and $\mathbb{Z}$.

In Murphy's textbook, where I've read almost all I know about Fredholm index, there is Atkinson's theorem, the fact, that Fredholm index is locally constant and $ind(ab)=ind(a)+ind(b)$. But in order to check the fact 1, I need also the fact, stated in the head of the question, or, equivalently, the following:

Fact 2. If $H$ is infinite-dimensional and separable then the set $\{a\in B(H)/K(H)\mid ind(a)=0\}$ is connected.

Where can I find the references? Where can I read that?