Metamath Proof Explorer


Theorem strlem3a

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. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypothesis strlem3a.1 𝑆 = ( 𝑥C ↦ ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) )
Assertion strlem3a ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → 𝑆 ∈ States )

Proof

Step Hyp Ref Expression
1 strlem3a.1 𝑆 = ( 𝑥C ↦ ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) )
2 id ( 𝑥C𝑥C )
3 simpl ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → 𝑢 ∈ ℋ )
4 pjhcl ( ( 𝑥C𝑢 ∈ ℋ ) → ( ( proj𝑥 ) ‘ 𝑢 ) ∈ ℋ )
5 2 3 4 syl2anr ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( ( proj𝑥 ) ‘ 𝑢 ) ∈ ℋ )
6 normcl ( ( ( proj𝑥 ) ‘ 𝑢 ) ∈ ℋ → ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∈ ℝ )
7 5 6 syl ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∈ ℝ )
8 7 resqcld ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ℝ )
9 7 sqge0d ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → 0 ≤ ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) )
10 normge0 ( ( ( proj𝑥 ) ‘ 𝑢 ) ∈ ℋ → 0 ≤ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) )
11 5 10 syl ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → 0 ≤ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) )
12 pjnorm ( ( 𝑥C𝑢 ∈ ℋ ) → ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ≤ ( norm𝑢 ) )
13 2 3 12 syl2anr ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ≤ ( norm𝑢 ) )
14 simplr ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( norm𝑢 ) = 1 )
15 13 14 breqtrd ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ≤ 1 )
16 2nn0 2 ∈ ℕ0
17 exple1 ( ( ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ∧ 0 ≤ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∧ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ≤ 1 ) ∧ 2 ∈ ℕ0 ) → ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 )
18 16 17 mpan2 ( ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ∧ 0 ≤ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ∧ ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ≤ 1 ) → ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 )
19 7 11 15 18 syl3anc ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 )
20 elicc01 ( ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∧ ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 ) )
21 8 9 19 20 syl3anbrc ( ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) ∧ 𝑥C ) → ( ( norm ‘ ( ( proj𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ( 0 [,] 1 ) )
22 21 1 fmptd ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → 𝑆 : C ⟶ ( 0 [,] 1 ) )
23 helch ℋ ∈ C
24 1 strlem2 ( ℋ ∈ C → ( 𝑆 ‘ ℋ ) = ( ( norm ‘ ( ( proj ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) )
25 23 24 ax-mp ( 𝑆 ‘ ℋ ) = ( ( norm ‘ ( ( proj ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 )
26 pjch1 ( 𝑢 ∈ ℋ → ( ( proj ‘ ℋ ) ‘ 𝑢 ) = 𝑢 )
27 26 fveq2d ( 𝑢 ∈ ℋ → ( norm ‘ ( ( proj ‘ ℋ ) ‘ 𝑢 ) ) = ( norm𝑢 ) )
28 27 oveq1d ( 𝑢 ∈ ℋ → ( ( norm ‘ ( ( proj ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( norm𝑢 ) ↑ 2 ) )
29 oveq1 ( ( norm𝑢 ) = 1 → ( ( norm𝑢 ) ↑ 2 ) = ( 1 ↑ 2 ) )
30 sq1 ( 1 ↑ 2 ) = 1
31 29 30 eqtrdi ( ( norm𝑢 ) = 1 → ( ( norm𝑢 ) ↑ 2 ) = 1 )
32 28 31 sylan9eq ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → ( ( norm ‘ ( ( proj ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) = 1 )
33 25 32 syl5eq ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → ( 𝑆 ‘ ℋ ) = 1 )
34 pjcjt2 ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) = ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) ) )
35 34 imp ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) = ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) )
36 35 fveq2d ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( norm ‘ ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) ) = ( norm ‘ ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) ) )
37 36 oveq1d ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( norm ‘ ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( norm ‘ ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) )
38 pjopyth ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( norm ‘ ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) )
39 38 imp ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( norm ‘ ( ( ( proj𝑧 ) ‘ 𝑢 ) + ( ( proj𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) )
40 37 39 eqtrd ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( norm ‘ ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) )
41 chjcl ( ( 𝑧C𝑤C ) → ( 𝑧 𝑤 ) ∈ C )
42 41 3adant3 ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) → ( 𝑧 𝑤 ) ∈ C )
43 42 adantr ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑧 𝑤 ) ∈ C )
44 1 strlem2 ( ( 𝑧 𝑤 ) ∈ C → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( norm ‘ ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) )
45 43 44 syl ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( norm ‘ ( ( proj ‘ ( 𝑧 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) )
46 3simpa ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) → ( 𝑧C𝑤C ) )
47 46 adantr ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑧C𝑤C ) )
48 1 strlem2 ( 𝑧C → ( 𝑆𝑧 ) = ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) )
49 1 strlem2 ( 𝑤C → ( 𝑆𝑤 ) = ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) )
50 48 49 oveqan12d ( ( 𝑧C𝑤C ) → ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) = ( ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) )
51 47 50 syl ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) = ( ( ( norm ‘ ( ( proj𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( norm ‘ ( ( proj𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) )
52 40 45 51 3eqtr4d ( ( ( 𝑧C𝑤C𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) )
53 52 3exp1 ( 𝑧C → ( 𝑤C → ( 𝑢 ∈ ℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) ) ) )
54 53 com3r ( 𝑢 ∈ ℋ → ( 𝑧C → ( 𝑤C → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) ) ) )
55 54 adantr ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → ( 𝑧C → ( 𝑤C → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) ) ) )
56 55 ralrimdv ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → ( 𝑧C → ∀ 𝑤C ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) ) )
57 56 ralrimiv ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → ∀ 𝑧C𝑤C ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) )
58 isst ( 𝑆 ∈ States ↔ ( 𝑆 : C ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑧C𝑤C ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 𝑤 ) ) = ( ( 𝑆𝑧 ) + ( 𝑆𝑤 ) ) ) ) )
59 22 33 57 58 syl3anbrc ( ( 𝑢 ∈ ℋ ∧ ( norm𝑢 ) = 1 ) → 𝑆 ∈ States )