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 = ( 𝑥 ∈ Cℋ ↦ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ) | |
| strlem3.2 | ⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) ) | ||
| strlem3.3 | ⊢ 𝐴 ∈ Cℋ | ||
| strlem3.4 | ⊢ 𝐵 ∈ Cℋ | ||
| Assertion | strlem3 | ⊢ ( 𝜑 → 𝑆 ∈ States ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | strlem3.1 | ⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ) | |
| 2 | strlem3.2 | ⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) ) | |
| 3 | strlem3.3 | ⊢ 𝐴 ∈ Cℋ | |
| 4 | strlem3.4 | ⊢ 𝐵 ∈ Cℋ | |
| 5 | eldifi | ⊢ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑢 ∈ 𝐴 ) | |
| 6 | 3 | cheli | ⊢ ( 𝑢 ∈ 𝐴 → 𝑢 ∈ ℋ ) |
| 7 | 5 6 | syl | ⊢ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑢 ∈ ℋ ) |
| 8 | 1 | strlem3a | ⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ States ) |
| 9 | 7 8 | sylan | ⊢ ( ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ States ) |
| 10 | 2 9 | sylbi | ⊢ ( 𝜑 → 𝑆 ∈ States ) |