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Theorem branmfn 31047
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 6847 . . 3 (𝐴 = 0 → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0)))
2 fveq2 6842 . . 3 (𝐴 = 0 → (norm𝐴) = (norm‘0))
31, 2eqeq12d 2752 . 2 (𝐴 = 0 → ((normfn‘(bra‘𝐴)) = (norm𝐴) ↔ (normfn‘(bra‘0)) = (norm‘0)))
4 brafn 30889 . . . . 5 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5 nmfnval 30818 . . . . 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 481 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8 nmfnsetre 30819 . . . . . . . 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 11199 . . . . . . 7 ℝ ⊆ ℝ*
119, 10sstrdi 3956 . . . . . 6 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12 normcl 30067 . . . . . . 7 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1312rexrd 11205 . . . . . 6 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ*)
1411, 13jca 512 . . . . 5 (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
1514adantr 481 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
16 vex 3449 . . . . . . . 8 𝑧 ∈ V
17 eqeq1 2740 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
1817anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
1918rexbidv 3175 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2016, 19elab 3630 . . . . . . 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 30886 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
2322fveq2d 6846 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2423adantr 481 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2521, 24sylan9eqr 2798 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26 bcs2 30124 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
27263expa 1118 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2827ancom1s 651 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2928adantr 481 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3025, 29eqbrtrd 5127 . . . . . . . . . . 11 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))
3130exp41 435 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((norm𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (norm𝐴)))))
3231imp4a 423 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))))
3332rexlimdv 3150 . . . . . . . 8 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴)))
3433imp 407 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (norm𝐴))
3520, 34sylan2b 594 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (norm𝐴))
3635ralrimiva 3143 . . . . 5 (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3736adantr 481 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3812recnd 11183 . . . . . . . . . 10 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℂ)
3938adantr 481 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
40 normne0 30072 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
4140biimpar 478 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
4239, 41reccld 11924 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
43 simpl 483 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
44 hvmulcl 29955 . . . . . . . . . . . 12 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4542, 43, 44syl2anc 584 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
46 norm1 30191 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
47 1le1 11783 . . . . . . . . . . . 12 1 ≤ 1
4846, 47eqbrtrdi 5144 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
49 ax-his3 30026 . . . . . . . . . . . . 13 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5042, 43, 43, 49syl3anc 1371 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5112adantr 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
5251, 41rereccld 11982 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
53 hiidrcl 30037 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
5453adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ·ih 𝐴) ∈ ℝ)
5552, 54remulcld 11185 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
5650, 55eqeltrd 2838 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) ∈ ℝ)
57 normgt0 30069 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
5857biimpa 477 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
5951, 58recgt0d 12089 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
60 0re 11157 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
61 ltle 11243 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6260, 61mpan 688 . . . . . . . . . . . . . . . 16 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6352, 59, 62sylc 65 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
64 hiidge0 30040 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
6564adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 ·ih 𝐴))
6652, 54, 63, 65mulge0d 11732 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
6766, 50breqtrrd 5133 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6856, 67absidd 15307 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6939, 41recid2d 11927 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm𝐴)) = 1)
7069oveq2d 7373 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((norm𝐴) · 1))
7139, 42, 39mul12d 11364 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))))
7238sqvald 14048 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = ((norm𝐴) · (norm𝐴)))
73 normsq 30076 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
7472, 73eqtr3d 2778 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7574adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7675oveq2d 7373 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7771, 76eqtrd 2776 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7838mulid1d 11172 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) · 1) = (norm𝐴))
7978adantr 481 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · 1) = (norm𝐴))
8070, 77, 793eqtr3rd 2785 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
8150, 68, 803eqtr4rd 2787 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
82 fveq2 6842 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (norm𝑦) = (norm‘((1 / (norm𝐴)) · 𝐴)))
8382breq1d 5115 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝑦) ≤ 1 ↔ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1))
84 fvoveq1 7380 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
8584eqeq2d 2747 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))))
8683, 85anbi12d 631 . . . . . . . . . . . 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 3581 . . . . . . . . . . 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 2747 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9089anbi2d 629 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9190rexbidva 3173 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9291adantr 481 . . . . . . . . . 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 256 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
94 eqeq1 2740 . . . . . . . . . . 11 (𝑥 = (norm𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
9594anbi2d 629 . . . . . . . . . 10 (𝑥 = (norm𝐴) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9695rexbidv 3175 . . . . . . . . 9 (𝑥 = (norm𝐴) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9739, 93, 96elabd 3633 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
98 breq2 5109 . . . . . . . . 9 (𝑤 = (norm𝐴) → (𝑧 < 𝑤𝑧 < (norm𝐴)))
9998rspcev 3581 . . . . . . . 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 580 . . . . . . 7 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
101100adantlr 713 . . . . . 6 ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
102101ex 413 . . . . 5 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
103102ralrimiva 3143 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
104 supxr2 13233 . . . 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 2776 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = (norm𝐴))
107 nmfn0 30929 . . . 4 (normfn‘( ℋ × {0})) = 0
108 bra0 30892 . . . . 5 (bra‘0) = ( ℋ × {0})
109108fveq2i 6845 . . . 4 (normfn‘(bra‘0)) = (normfn‘( ℋ × {0}))
110 norm0 30070 . . . 4 (norm‘0) = 0
111107, 109, 1103eqtr4i 2774 . . 3 (normfn‘(bra‘0)) = (norm‘0)
112111a1i 11 . 2 (𝐴 ∈ ℋ → (normfn‘(bra‘0)) = (norm‘0))
1133, 106, 112pm2.61ne 3030 1 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  wss 3910  {csn 4586   class class class wbr 5105   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   · cmul 11056  *cxr 11188   < clt 11189  cle 11190   / cdiv 11812  2c2 12208  cexp 13967  abscabs 15119  chba 29861   · csm 29863   ·ih csp 29864  normcno 29865  0c0v 29866  normfncnmf 29893  bracbr 29898
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131  ax-hilex 29941  ax-hfvadd 29942  ax-hvcom 29943  ax-hvass 29944  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmulass 29949  ax-hvdistr1 29950  ax-hvdistr2 29951  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his2 30025  ax-his3 30026  ax-his4 30027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-grpo 29435  df-gid 29436  df-ginv 29437  df-gdiv 29438  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-vs 29541  df-nmcv 29542  df-ims 29543  df-dip 29643  df-ph 29755  df-hnorm 29910  df-hba 29911  df-hvsub 29913  df-nmfn 30787  df-lnfn 30790  df-bra 30792
This theorem is referenced by:  brabn  31048
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