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Theorem branmfn 29891
Description: The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
branmfn (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))

Proof of Theorem branmfn
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6654 . . 3 (𝐴 = 0 → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0)))
2 fveq2 6649 . . 3 (𝐴 = 0 → (norm𝐴) = (norm‘0))
31, 2eqeq12d 2817 . 2 (𝐴 = 0 → ((normfn‘(bra‘𝐴)) = (norm𝐴) ↔ (normfn‘(bra‘0)) = (norm‘0)))
4 brafn 29733 . . . . 5 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5 nmfnval 29662 . . . . 5 ((bra‘𝐴): ℋ⟶ℂ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
64, 5syl 17 . . . 4 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
76adantr 484 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8 nmfnsetre 29663 . . . . . . . 8 ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
94, 8syl 17 . . . . . . 7 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
10 ressxr 10678 . . . . . . 7 ℝ ⊆ ℝ*
119, 10sstrdi 3930 . . . . . 6 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12 normcl 28911 . . . . . . 7 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1312rexrd 10684 . . . . . 6 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ*)
1411, 13jca 515 . . . . 5 (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
1514adantr 484 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
16 vex 3447 . . . . . . . 8 𝑧 ∈ V
17 eqeq1 2805 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
1817anbi2d 631 . . . . . . . . 9 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
1918rexbidv 3259 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2016, 19elab 3618 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
21 id 22 . . . . . . . . . . . . 13 (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
22 braval 29730 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
2322fveq2d 6653 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2423adantr 484 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2521, 24sylan9eqr 2858 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26 bcs2 28968 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
27263expa 1115 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2827ancom1s 652 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2928adantr 484 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3025, 29eqbrtrd 5055 . . . . . . . . . . 11 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))
3130exp41 438 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((norm𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (norm𝐴)))))
3231imp4a 426 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))))
3332rexlimdv 3245 . . . . . . . 8 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴)))
3433imp 410 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (norm𝐴))
3520, 34sylan2b 596 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (norm𝐴))
3635ralrimiva 3152 . . . . 5 (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3736adantr 484 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3812recnd 10662 . . . . . . . . . 10 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℂ)
3938adantr 484 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
40 normne0 28916 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
4140biimpar 481 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
4239, 41reccld 11402 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
43 simpl 486 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
44 hvmulcl 28799 . . . . . . . . . . . 12 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4542, 43, 44syl2anc 587 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
46 norm1 29035 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
47 1le1 11261 . . . . . . . . . . . 12 1 ≤ 1
4846, 47eqbrtrdi 5072 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
49 ax-his3 28870 . . . . . . . . . . . . 13 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5042, 43, 43, 49syl3anc 1368 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5112adantr 484 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
5251, 41rereccld 11460 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
53 hiidrcl 28881 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
5453adantr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ·ih 𝐴) ∈ ℝ)
5552, 54remulcld 10664 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
5650, 55eqeltrd 2893 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) ∈ ℝ)
57 normgt0 28913 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
5857biimpa 480 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
5951, 58recgt0d 11567 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
60 0re 10636 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
61 ltle 10722 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6260, 61mpan 689 . . . . . . . . . . . . . . . 16 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6352, 59, 62sylc 65 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
64 hiidge0 28884 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
6564adantr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 ·ih 𝐴))
6652, 54, 63, 65mulge0d 11210 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
6766, 50breqtrrd 5061 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6856, 67absidd 14777 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6939, 41recid2d 11405 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm𝐴)) = 1)
7069oveq2d 7155 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((norm𝐴) · 1))
7139, 42, 39mul12d 10842 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))))
7238sqvald 13507 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = ((norm𝐴) · (norm𝐴)))
73 normsq 28920 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
7472, 73eqtr3d 2838 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7574adantr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7675oveq2d 7155 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7771, 76eqtrd 2836 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7838mulid1d 10651 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) · 1) = (norm𝐴))
7978adantr 484 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · 1) = (norm𝐴))
8070, 77, 793eqtr3rd 2845 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
8150, 68, 803eqtr4rd 2847 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
82 fveq2 6649 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (norm𝑦) = (norm‘((1 / (norm𝐴)) · 𝐴)))
8382breq1d 5043 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝑦) ≤ 1 ↔ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1))
84 fvoveq1 7162 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
8584eqeq2d 2812 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))))
8683, 85anbi12d 633 . . . . . . . . . . . 12 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))) ↔ ((norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1 ∧ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))))
8786rspcev 3574 . . . . . . . . . . 11 ((((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))))
8845, 48, 81, 87syl12anc 835 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
8923eqeq2d 2812 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9089anbi2d 631 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9190rexbidva 3258 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9291adantr 484 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9388, 92mpbird 260 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
94 eqeq1 2805 . . . . . . . . . . 11 (𝑥 = (norm𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
9594anbi2d 631 . . . . . . . . . 10 (𝑥 = (norm𝐴) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9695rexbidv 3259 . . . . . . . . 9 (𝑥 = (norm𝐴) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9739, 93, 96elabd 3620 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
98 breq2 5037 . . . . . . . . 9 (𝑤 = (norm𝐴) → (𝑧 < 𝑤𝑧 < (norm𝐴)))
9998rspcev 3574 . . . . . . . 8 (((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)
10097, 99sylan 583 . . . . . . 7 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
101100adantlr 714 . . . . . 6 ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
102101ex 416 . . . . 5 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
103102ralrimiva 3152 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
104 supxr2 12699 . . . 4 ((({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))
10515, 37, 103, 104syl12anc 835 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (norm𝐴))
1067, 105eqtrd 2836 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = (norm𝐴))
107 nmfn0 29773 . . . 4 (normfn‘( ℋ × {0})) = 0
108 bra0 29736 . . . . 5 (bra‘0) = ( ℋ × {0})
109108fveq2i 6652 . . . 4 (normfn‘(bra‘0)) = (normfn‘( ℋ × {0}))
110 norm0 28914 . . . 4 (norm‘0) = 0
111107, 109, 1103eqtr4i 2834 . . 3 (normfn‘(bra‘0)) = (norm‘0)
112111a1i 11 . 2 (𝐴 ∈ ℋ → (normfn‘(bra‘0)) = (norm‘0))
1133, 106, 112pm2.61ne 3075 1 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  wss 3884  {csn 4528   class class class wbr 5033   × cxp 5521  wf 6324  cfv 6328  (class class class)co 7139  supcsup 8892  cc 10528  cr 10529  0cc0 10530  1c1 10531   · cmul 10535  *cxr 10667   < clt 10668  cle 10669   / cdiv 11290  2c2 11684  cexp 13429  abscabs 14588  chba 28705   · csm 28707   ·ih csp 28708  normcno 28709  0c0v 28710  normfncnmf 28737  bracbr 28742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610  ax-hilex 28785  ax-hfvadd 28786  ax-hvcom 28787  ax-hvass 28788  ax-hv0cl 28789  ax-hvaddid 28790  ax-hfvmul 28791  ax-hvmulid 28792  ax-hvmulass 28793  ax-hvdistr1 28794  ax-hvdistr2 28795  ax-hvmul0 28796  ax-hfi 28865  ax-his1 28868  ax-his2 28869  ax-his3 28870  ax-his4 28871
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-icc 12737  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-cnfld 20095  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-cn 21835  df-cnp 21836  df-t1 21922  df-haus 21923  df-tx 22170  df-hmeo 22363  df-xms 22930  df-ms 22931  df-tms 22932  df-grpo 28279  df-gid 28280  df-ginv 28281  df-gdiv 28282  df-ablo 28331  df-vc 28345  df-nv 28378  df-va 28381  df-ba 28382  df-sm 28383  df-0v 28384  df-vs 28385  df-nmcv 28386  df-ims 28387  df-dip 28487  df-ph 28599  df-hnorm 28754  df-hba 28755  df-hvsub 28757  df-nmfn 29631  df-lnfn 29634  df-bra 29636
This theorem is referenced by:  brabn  29892
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