Description: Lemma for strong set of CH states 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, 30-Jun-2006) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | hstrlem3.1 | ⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) | |
| hstrlem3.2 | ⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) ) | ||
| hstrlem3.3 | ⊢ 𝐴 ∈ Cℋ | ||
| hstrlem3.4 | ⊢ 𝐵 ∈ Cℋ | ||
| Assertion | hstrlem3 | ⊢ ( 𝜑 → 𝑆 ∈ CHStates ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hstrlem3.1 | ⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) | |
| 2 | hstrlem3.2 | ⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) ) | |
| 3 | hstrlem3.3 | ⊢ 𝐴 ∈ Cℋ | |
| 4 | hstrlem3.4 | ⊢ 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 | hstrlem3a | ⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ CHStates ) |
| 9 | 7 8 | sylan | ⊢ ( ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ CHStates ) |
| 10 | 2 9 | sylbi | ⊢ ( 𝜑 → 𝑆 ∈ CHStates ) |