Metamath Proof Explorer

Theorem hstle

Description: Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hstle ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 hstnmoc ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) = 1 )
2 1 adantlr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) = 1 )
3 2 oveq2d ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) )
4 hstcl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ )
5 normcl ⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ∈ ℝ )
6 4 5 syl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ∈ ℝ )
7 6 resqcld ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ∈ ℝ )
8 7 adantr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ∈ ℝ )
9 8 recnd ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ∈ ℂ )
10 hstcl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ 𝐵 ) ∈ ℋ )
11 normcl ⊢ ( ( 𝑆 ‘ 𝐵 ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ )
12 10 11 syl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ )
13 12 resqcld ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ )
14 13 adantlr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ )
15 14 recnd ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℂ )
16 choccl ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
17 hstcl ⊢ ( ( 𝑆 ∈ CHStates ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ∈ ℋ )
18 16 17 sylan2 ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ∈ ℋ )
19 normcl ⊢ ( ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℝ )
20 18 19 syl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℝ )
21 20 resqcld ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
22 21 adantlr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
23 22 recnd ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ∈ ℂ )
24 9 15 23 add12d ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) )
25 3 24 eqtr3d ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) )
26 25 adantrr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) )
27 16 adantr ⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
28 ococ ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
29 28 sseq2d ⊢ ( 𝐵 ∈ Cℋ → ( 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ↔ 𝐴 ⊆ 𝐵 ) )
30 29 biimpar ⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) )
31 27 30 jca ⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
32 hstpyth ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) )
33 31 32 sylan2 ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) )
34 chjcl ⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ )
35 16 34 sylan2 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ )
36 hstcl ⊢ ( ( 𝑆 ∈ CHStates ∧ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℋ )
37 35 36 sylan2 ⊢ ( ( 𝑆 ∈ CHStates ∧ ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℋ )
38 37 anassrs ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℋ )
39 normcl ⊢ ( ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ∈ ℝ )
40 38 39 syl ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ∈ ℝ )
41 normge0 ⊢ ( ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ∈ ℋ → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) )
42 38 41 syl ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) )
43 hstle1 ⊢ ( ( 𝑆 ∈ CHStates ∧ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ≤ 1 )
44 35 43 sylan2 ⊢ ( ( 𝑆 ∈ CHStates ∧ ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ) → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ≤ 1 )
45 44 anassrs ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ≤ 1 )
46 1re ⊢ 1 ∈ ℝ
47 le2sq2 ⊢ ( ( ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ) ∧ ( 1 ∈ ℝ ∧ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ≤ 1 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) )
48 46 47 mpanr1 ⊢ ( ( ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ) ∧ ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ≤ 1 ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) )
49 40 42 45 48 syl21anc ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) )
50 sq1 ⊢ ( 1 ↑ 2 ) = 1
51 49 50 breqtrdi ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) ≤ 1 )
52 51 adantrr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) ) ↑ 2 ) ≤ 1 )
53 33 52 eqbrtrrd ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ≤ 1 )
54 8 22 readdcld ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ∈ ℝ )
55 leadd2 ⊢ ( ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ∈ ℝ ∧ 1 ∈ ℝ ∧ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ ) → ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ≤ 1 ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
56 46 55 mp3an2 ⊢ ( ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ∈ ℝ ∧ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ ) → ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ≤ 1 ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
57 54 14 56 syl2anc ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ≤ 1 ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
58 57 adantrr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ≤ 1 ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
59 53 58 mpbid ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ ( ⊥ ‘ 𝐵 ) ) ) ↑ 2 ) ) ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) )
60 26 59 eqbrtrd ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) )
61 leadd1 ⊢ ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ∈ ℝ ∧ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
62 46 61 mp3an3 ⊢ ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ∈ ℝ ∧ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℝ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
63 8 14 62 syl2anc ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
64 63 adantrr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ↔ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + 1 ) ≤ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) + 1 ) ) )
65 60 64 mpbird ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) )
66 normge0 ⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) )
67 4 66 syl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) )
68 6 67 jca ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ) )
69 68 adantr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ) )
70 normge0 ⊢ ( ( 𝑆 ‘ 𝐵 ) ∈ ℋ → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) )
71 10 70 syl ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) )
72 12 71 jca ⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) )
73 72 adantlr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) )
74 le2sq ⊢ ( ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ) ∧ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) )
75 69 73 74 syl2anc ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) )
76 75 adantrr ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) ≤ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) )
77 65 76 mpbird ⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) )