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Theorem branmfn 28348
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 fveq2 6103 . . . 4 (𝐴 = 0 → (bra‘𝐴) = (bra‘0))
21fveq2d 6107 . . 3 (𝐴 = 0 → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0)))
3 fveq2 6103 . . 3 (𝐴 = 0 → (norm𝐴) = (norm‘0))
42, 3eqeq12d 2625 . 2 (𝐴 = 0 → ((normfn‘(bra‘𝐴)) = (norm𝐴) ↔ (normfn‘(bra‘0)) = (norm‘0)))
5 brafn 28190 . . . . 5 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
6 nmfnval 28119 . . . . 5 ((bra‘𝐴): ℋ⟶ℂ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
75, 6syl 17 . . . 4 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
87adantr 480 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
9 nmfnsetre 28120 . . . . . . . 8 ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
105, 9syl 17 . . . . . . 7 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
11 ressxr 9962 . . . . . . 7 ℝ ⊆ ℝ*
1210, 11syl6ss 3580 . . . . . 6 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
13 normcl 27366 . . . . . . 7 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1413rexrd 9968 . . . . . 6 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ*)
1512, 14jca 553 . . . . 5 (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
1615adantr 480 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
17 vex 3176 . . . . . . . 8 𝑧 ∈ V
18 eqeq1 2614 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
1918anbi2d 736 . . . . . . . . 9 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2019rexbidv 3034 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2117, 20elab 3319 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
22 id 22 . . . . . . . . . . . . 13 (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
23 braval 28187 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
2423fveq2d 6107 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2524adantr 480 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2622, 25sylan9eqr 2666 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
27 bcs2 27423 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
28273expa 1257 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2928ancom1s 843 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3029adantr 480 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3126, 30eqbrtrd 4605 . . . . . . . . . . 11 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))
3231exp41 636 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((norm𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (norm𝐴)))))
3332imp4a 612 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))))
3433rexlimdv 3012 . . . . . . . 8 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴)))
3534imp 444 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (norm𝐴))
3621, 35sylan2b 491 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (norm𝐴))
3736ralrimiva 2949 . . . . 5 (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3837adantr 480 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3913recnd 9947 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℂ)
4039adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
41 normne0 27371 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
4241biimpar 501 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
4340, 42reccld 10673 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
44 simpl 472 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
45 hvmulcl 27254 . . . . . . . . . . . 12 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4643, 44, 45syl2anc 691 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
47 norm1 27490 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
48 1le1 10534 . . . . . . . . . . . 12 1 ≤ 1
4947, 48syl6eqbr 4622 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
50 ax-his3 27325 . . . . . . . . . . . . 13 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5143, 44, 44, 50syl3anc 1318 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5213adantr 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
5352, 42rereccld 10731 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
54 hiidrcl 27336 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
5554adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ·ih 𝐴) ∈ ℝ)
5653, 55remulcld 9949 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
5751, 56eqeltrd 2688 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) ∈ ℝ)
58 normgt0 27368 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
5958biimpa 500 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
6052, 59recgt0d 10837 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
61 0re 9919 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
62 ltle 10005 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6361, 62mpan 702 . . . . . . . . . . . . . . . 16 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6453, 60, 63sylc 63 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
65 hiidge0 27339 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
6665adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 ·ih 𝐴))
6753, 55, 64, 66mulge0d 10483 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
6867, 51breqtrrd 4611 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6957, 68absidd 14009 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
7040, 42recid2d 10676 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm𝐴)) = 1)
7170oveq2d 6565 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((norm𝐴) · 1))
7240, 43, 40mul12d 10124 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))))
7339sqvald 12867 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = ((norm𝐴) · (norm𝐴)))
74 normsq 27375 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
7573, 74eqtr3d 2646 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7675adantr 480 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7776oveq2d 6565 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7872, 77eqtrd 2644 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7939mulid1d 9936 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) · 1) = (norm𝐴))
8079adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · 1) = (norm𝐴))
8171, 78, 803eqtr3rd 2653 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
8251, 69, 813eqtr4rd 2655 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
83 fveq2 6103 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (norm𝑦) = (norm‘((1 / (norm𝐴)) · 𝐴)))
8483breq1d 4593 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝑦) ≤ 1 ↔ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1))
85 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (𝑦 ·ih 𝐴) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
8685fveq2d 6107 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
8786eqeq2d 2620 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))))
8884, 87anbi12d 743 . . . . . . . . . . . 12 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))) ↔ ((norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1 ∧ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))))
8988rspcev 3282 . . . . . . . . . . 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))))
9046, 49, 82, 89syl12anc 1316 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9124eqeq2d 2620 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9291anbi2d 736 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9392rexbidva 3031 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9493adantr 480 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9590, 94mpbird 246 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
96 fvex 6113 . . . . . . . . . 10 (norm𝐴) ∈ V
97 eqeq1 2614 . . . . . . . . . . . 12 (𝑥 = (norm𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
9897anbi2d 736 . . . . . . . . . . 11 (𝑥 = (norm𝐴) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9998rexbidv 3034 . . . . . . . . . 10 (𝑥 = (norm𝐴) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
10096, 99elab 3319 . . . . . . . . 9 ((norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
10195, 100sylibr 223 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
102 breq2 4587 . . . . . . . . 9 (𝑤 = (norm𝐴) → (𝑧 < 𝑤𝑧 < (norm𝐴)))
103102rspcev 3282 . . . . . . . 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)
104101, 103sylan 487 . . . . . . 7 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
105104adantlr 747 . . . . . 6 ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
106105ex 449 . . . . 5 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
107106ralrimiva 2949 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
108 supxr2 12016 . . . 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))
10916, 38, 107, 108syl12anc 1316 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (norm𝐴))
1108, 109eqtrd 2644 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = (norm𝐴))
111 nmfn0 28230 . . . 4 (normfn‘( ℋ × {0})) = 0
112 bra0 28193 . . . . 5 (bra‘0) = ( ℋ × {0})
113112fveq2i 6106 . . . 4 (normfn‘(bra‘0)) = (normfn‘( ℋ × {0}))
114 norm0 27369 . . . 4 (norm‘0) = 0
115111, 113, 1143eqtr4i 2642 . . 3 (normfn‘(bra‘0)) = (norm‘0)
116115a1i 11 . 2 (𝐴 ∈ ℋ → (normfn‘(bra‘0)) = (norm‘0))
1174, 110, 116pm2.61ne 2867 1 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wne 2780  wral 2896  wrex 2897  wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  supcsup 8229  cc 9813  cr 9814  0cc0 9815  1c1 9816   · cmul 9820  *cxr 9952   < clt 9953  cle 9954   / cdiv 10563  2c2 10947  cexp 12722  abscabs 13822  chil 27160   · csm 27162   ·ih csp 27163  normcno 27164  0c0v 27165  normfncnmf 27192  bracbr 27197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895  ax-hilex 27240  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmulass 27248  ax-hvdistr1 27249  ax-hvdistr2 27250  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his2 27324  ax-his3 27325  ax-his4 27326
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840  df-dip 26940  df-ph 27052  df-hnorm 27209  df-hba 27210  df-hvsub 27212  df-nmfn 28088  df-lnfn 28091  df-bra 28093
This theorem is referenced by:  brabn  28349
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