Theorem strlem3

Description: Lemma for strong state theorem: the function S , that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	strlem3.1	$$⊢S=\left(x\in {\mathbf{C}}_{\mathscr{H}}⟼{{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(x\right)\left(u\right)\right)}^2\right)$$
	strlem3.2	$$⊢\phi \leftrightarrow \left(u\in \left(A\setminus B\right)\wedge {norm}_{ℎ}\left(u\right)=1\right)$$
	strlem3.3	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	strlem3.4	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	strlem3	$$⊢\phi \to S\in \mathrm{States}$$

Proof

Step	Hyp	Ref	Expression
1		strlem3.1	$$⊢S=\left(x\in {\mathbf{C}}_{\mathscr{H}}⟼{{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(x\right)\left(u\right)\right)}^2\right)$$
2		strlem3.2	$$⊢\phi \leftrightarrow \left(u\in \left(A\setminus B\right)\wedge {norm}_{ℎ}\left(u\right)=1\right)$$
3		strlem3.3	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
4		strlem3.4	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
5		eldifi	$$⊢u\in \left(A\setminus B\right)\to u\in A$$
6	3	cheli	$$⊢u\in A\to u\in \mathscr{H}$$
7	5 6	syl	$$⊢u\in \left(A\setminus B\right)\to u\in \mathscr{H}$$
8	1	strlem3a	$$⊢\left(u\in \mathscr{H}\wedge {norm}_{ℎ}\left(u\right)=1\right)\to S\in \mathrm{States}$$
9	7 8	sylan	$$⊢\left(u\in \left(A\setminus B\right)\wedge {norm}_{ℎ}\left(u\right)=1\right)\to S\in \mathrm{States}$$
10	2 9	sylbi	$$⊢\phi \to S\in \mathrm{States}$$