Metamath Proof Explorer


Theorem branmfn

Description: The norm of the bra function. (Contributed by NM, 24-May-2006) (New usage is discouraged.)

Ref Expression
Assertion branmfn ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( norm𝐴 ) )

Proof

Step Hyp Ref Expression
1 2fveq3 ( 𝐴 = 0 → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( normfn ‘ ( bra ‘ 0 ) ) )
2 fveq2 ( 𝐴 = 0 → ( norm𝐴 ) = ( norm ‘ 0 ) )
3 1 2 eqeq12d ( 𝐴 = 0 → ( ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( norm𝐴 ) ↔ ( normfn ‘ ( bra ‘ 0 ) ) = ( norm ‘ 0 ) ) )
4 brafn ( 𝐴 ∈ ℋ → ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ )
5 nmfnval ( ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) )
6 4 5 syl ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) )
7 6 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( normfn ‘ ( bra ‘ 𝐴 ) ) = sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) )
8 nmfnsetre ( ( bra ‘ 𝐴 ) : ℋ ⟶ ℂ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ )
9 4 8 syl ( 𝐴 ∈ ℋ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ )
10 ressxr ℝ ⊆ ℝ*
11 9 10 sstrdi ( 𝐴 ∈ ℋ → { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* )
12 normcl ( 𝐴 ∈ ℋ → ( norm𝐴 ) ∈ ℝ )
13 12 rexrd ( 𝐴 ∈ ℋ → ( norm𝐴 ) ∈ ℝ* )
14 11 13 jca ( 𝐴 ∈ ℋ → ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( norm𝐴 ) ∈ ℝ* ) )
15 14 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( norm𝐴 ) ∈ ℝ* ) )
16 vex 𝑧 ∈ V
17 eqeq1 ( 𝑥 = 𝑧 → ( 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) )
18 17 anbi2d ( 𝑥 = 𝑧 → ( ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) )
19 18 rexbidv ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) )
20 16 19 elab ( 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ↔ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) )
21 id ( 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) → 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) )
22 braval ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) = ( 𝑦 ·ih 𝐴 ) )
23 22 fveq2d ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) )
24 23 adantr ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) → ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) )
25 21 24 sylan9eqr ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) )
26 bcs2 ( ( 𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ ( norm𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( norm𝐴 ) )
27 26 3expa ( ( ( 𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( norm𝐴 ) )
28 27 ancom1s ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( norm𝐴 ) )
29 28 adantr ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ≤ ( norm𝐴 ) )
30 25 29 eqbrtrd ( ( ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( norm𝑦 ) ≤ 1 ) ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( norm𝐴 ) )
31 30 exp41 ( 𝐴 ∈ ℋ → ( 𝑦 ∈ ℋ → ( ( norm𝑦 ) ≤ 1 → ( 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) → 𝑧 ≤ ( norm𝐴 ) ) ) ) )
32 31 imp4a ( 𝐴 ∈ ℋ → ( 𝑦 ∈ ℋ → ( ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( norm𝐴 ) ) ) )
33 32 rexlimdv ( 𝐴 ∈ ℋ → ( ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) → 𝑧 ≤ ( norm𝐴 ) ) )
34 33 imp ( ( 𝐴 ∈ ℋ ∧ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑧 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) → 𝑧 ≤ ( norm𝐴 ) )
35 20 34 sylan2b ( ( 𝐴 ∈ ℋ ∧ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ) → 𝑧 ≤ ( norm𝐴 ) )
36 35 ralrimiva ( 𝐴 ∈ ℋ → ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( norm𝐴 ) )
37 36 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( norm𝐴 ) )
38 12 recnd ( 𝐴 ∈ ℋ → ( norm𝐴 ) ∈ ℂ )
39 38 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) ∈ ℂ )
40 normne0 ( 𝐴 ∈ ℋ → ( ( norm𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )
41 40 biimpar ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) ≠ 0 )
42 39 41 reccld ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( 1 / ( norm𝐴 ) ) ∈ ℂ )
43 simpl ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℋ )
44 hvmulcl ( ( ( 1 / ( norm𝐴 ) ) ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ∈ ℋ )
45 42 43 44 syl2anc ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ∈ ℋ )
46 norm1 ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) = 1 )
47 1le1 1 ≤ 1
48 46 47 eqbrtrdi ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) ≤ 1 )
49 ax-his3 ( ( ( 1 / ( norm𝐴 ) ) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) = ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
50 42 43 43 49 syl3anc ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) = ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
51 12 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) ∈ ℝ )
52 51 41 rereccld ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( 1 / ( norm𝐴 ) ) ∈ ℝ )
53 hiidrcl ( 𝐴 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )
54 53 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )
55 52 54 remulcld ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) ∈ ℝ )
56 50 55 eqeltrd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ∈ ℝ )
57 normgt0 ( 𝐴 ∈ ℋ → ( 𝐴 ≠ 0 ↔ 0 < ( norm𝐴 ) ) )
58 57 biimpa ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 < ( norm𝐴 ) )
59 51 58 recgt0d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 < ( 1 / ( norm𝐴 ) ) )
60 0re 0 ∈ ℝ
61 ltle ( ( 0 ∈ ℝ ∧ ( 1 / ( norm𝐴 ) ) ∈ ℝ ) → ( 0 < ( 1 / ( norm𝐴 ) ) → 0 ≤ ( 1 / ( norm𝐴 ) ) ) )
62 60 61 mpan ( ( 1 / ( norm𝐴 ) ) ∈ ℝ → ( 0 < ( 1 / ( norm𝐴 ) ) → 0 ≤ ( 1 / ( norm𝐴 ) ) ) )
63 52 59 62 sylc ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 ≤ ( 1 / ( norm𝐴 ) ) )
64 hiidge0 ( 𝐴 ∈ ℋ → 0 ≤ ( 𝐴 ·ih 𝐴 ) )
65 64 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 ≤ ( 𝐴 ·ih 𝐴 ) )
66 52 54 63 65 mulge0d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 ≤ ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
67 66 50 breqtrrd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → 0 ≤ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) )
68 56 67 absidd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) = ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) )
69 39 41 recid2d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( 1 / ( norm𝐴 ) ) · ( norm𝐴 ) ) = 1 )
70 69 oveq2d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( norm𝐴 ) · ( ( 1 / ( norm𝐴 ) ) · ( norm𝐴 ) ) ) = ( ( norm𝐴 ) · 1 ) )
71 39 42 39 mul12d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( norm𝐴 ) · ( ( 1 / ( norm𝐴 ) ) · ( norm𝐴 ) ) ) = ( ( 1 / ( norm𝐴 ) ) · ( ( norm𝐴 ) · ( norm𝐴 ) ) ) )
72 38 sqvald ( 𝐴 ∈ ℋ → ( ( norm𝐴 ) ↑ 2 ) = ( ( norm𝐴 ) · ( norm𝐴 ) ) )
73 normsq ( 𝐴 ∈ ℋ → ( ( norm𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) )
74 72 73 eqtr3d ( 𝐴 ∈ ℋ → ( ( norm𝐴 ) · ( norm𝐴 ) ) = ( 𝐴 ·ih 𝐴 ) )
75 74 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( norm𝐴 ) · ( norm𝐴 ) ) = ( 𝐴 ·ih 𝐴 ) )
76 75 oveq2d ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( 1 / ( norm𝐴 ) ) · ( ( norm𝐴 ) · ( norm𝐴 ) ) ) = ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
77 71 76 eqtrd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( norm𝐴 ) · ( ( 1 / ( norm𝐴 ) ) · ( norm𝐴 ) ) ) = ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
78 38 mulid1d ( 𝐴 ∈ ℋ → ( ( norm𝐴 ) · 1 ) = ( norm𝐴 ) )
79 78 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ( norm𝐴 ) · 1 ) = ( norm𝐴 ) )
80 70 77 79 3eqtr3rd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) = ( ( 1 / ( norm𝐴 ) ) · ( 𝐴 ·ih 𝐴 ) ) )
81 50 68 80 3eqtr4rd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) = ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) )
82 fveq2 ( 𝑦 = ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) → ( norm𝑦 ) = ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) )
83 82 breq1d ( 𝑦 = ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) → ( ( norm𝑦 ) ≤ 1 ↔ ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) ≤ 1 ) )
84 fvoveq1 ( 𝑦 = ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) → ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) = ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) )
85 84 eqeq2d ( 𝑦 = ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) → ( ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ↔ ( norm𝐴 ) = ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) ) )
86 83 85 anbi12d ( 𝑦 = ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) → ( ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ↔ ( ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) ) ) )
87 86 rspcev ( ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ∈ ℋ ∧ ( ( norm ‘ ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( ( 1 / ( norm𝐴 ) ) · 𝐴 ) ·ih 𝐴 ) ) ) ) → ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) )
88 45 48 81 87 syl12anc ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) )
89 23 eqeq2d ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) )
90 89 anbi2d ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) )
91 90 rexbidva ( 𝐴 ∈ ℋ → ( ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) )
92 91 adantr ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( 𝑦 ·ih 𝐴 ) ) ) ) )
93 88 92 mpbird ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) )
94 eqeq1 ( 𝑥 = ( norm𝐴 ) → ( 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ↔ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) )
95 94 anbi2d ( 𝑥 = ( norm𝐴 ) → ( ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) )
96 95 rexbidv ( 𝑥 = ( norm𝐴 ) → ( ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ↔ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ ( norm𝐴 ) = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) ) )
97 39 93 96 elabd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( norm𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } )
98 breq2 ( 𝑤 = ( norm𝐴 ) → ( 𝑧 < 𝑤𝑧 < ( norm𝐴 ) ) )
99 98 rspcev ( ( ( norm𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ∧ 𝑧 < ( norm𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 )
100 97 99 sylan ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) ∧ 𝑧 < ( norm𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 )
101 100 adantlr ( ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( norm𝐴 ) ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 )
102 101 ex ( ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) ∧ 𝑧 ∈ ℝ ) → ( 𝑧 < ( norm𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) )
103 102 ralrimiva ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ∀ 𝑧 ∈ ℝ ( 𝑧 < ( norm𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) )
104 supxr2 ( ( ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } ⊆ ℝ* ∧ ( norm𝐴 ) ∈ ℝ* ) ∧ ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 ≤ ( norm𝐴 ) ∧ ∀ 𝑧 ∈ ℝ ( 𝑧 < ( norm𝐴 ) → ∃ 𝑤 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } 𝑧 < 𝑤 ) ) ) → sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) = ( norm𝐴 ) )
105 15 37 103 104 syl12anc ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → sup ( { 𝑥 ∣ ∃ 𝑦 ∈ ℋ ( ( norm𝑦 ) ≤ 1 ∧ 𝑥 = ( abs ‘ ( ( bra ‘ 𝐴 ) ‘ 𝑦 ) ) ) } , ℝ* , < ) = ( norm𝐴 ) )
106 7 105 eqtrd ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ) → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( norm𝐴 ) )
107 nmfn0 ( normfn ‘ ( ℋ × { 0 } ) ) = 0
108 bra0 ( bra ‘ 0 ) = ( ℋ × { 0 } )
109 108 fveq2i ( normfn ‘ ( bra ‘ 0 ) ) = ( normfn ‘ ( ℋ × { 0 } ) )
110 norm0 ( norm ‘ 0 ) = 0
111 107 109 110 3eqtr4i ( normfn ‘ ( bra ‘ 0 ) ) = ( norm ‘ 0 )
112 111 a1i ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 0 ) ) = ( norm ‘ 0 ) )
113 3 106 112 pm2.61ne ( 𝐴 ∈ ℋ → ( normfn ‘ ( bra ‘ 𝐴 ) ) = ( norm𝐴 ) )