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Theorem branmfn 28934
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 6178 . . . 4 (𝐴 = 0 → (bra‘𝐴) = (bra‘0))
21fveq2d 6182 . . 3 (𝐴 = 0 → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0)))
3 fveq2 6178 . . 3 (𝐴 = 0 → (norm𝐴) = (norm‘0))
42, 3eqeq12d 2635 . 2 (𝐴 = 0 → ((normfn‘(bra‘𝐴)) = (norm𝐴) ↔ (normfn‘(bra‘0)) = (norm‘0)))
5 brafn 28776 . . . . 5 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
6 nmfnval 28705 . . . . 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 481 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
9 nmfnsetre 28706 . . . . . . . 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 10068 . . . . . . 7 ℝ ⊆ ℝ*
1210, 11syl6ss 3607 . . . . . 6 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
13 normcl 27952 . . . . . . 7 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1413rexrd 10074 . . . . . 6 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ*)
1512, 14jca 554 . . . . 5 (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
1615adantr 481 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
17 vex 3198 . . . . . . . 8 𝑧 ∈ V
18 eqeq1 2624 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
1918anbi2d 739 . . . . . . . . 9 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2019rexbidv 3048 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2117, 20elab 3344 . . . . . . 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 28773 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
2423fveq2d 6182 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2524adantr 481 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2622, 25sylan9eqr 2676 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
27 bcs2 28009 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
28273expa 1263 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2928ancom1s 846 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3029adantr 481 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3126, 30eqbrtrd 4666 . . . . . . . . . . 11 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))
3231exp41 637 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((norm𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (norm𝐴)))))
3332imp4a 613 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))))
3433rexlimdv 3026 . . . . . . . 8 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴)))
3534imp 445 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (norm𝐴))
3621, 35sylan2b 492 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (norm𝐴))
3736ralrimiva 2963 . . . . 5 (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3837adantr 481 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3913recnd 10053 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℂ)
4039adantr 481 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
41 normne0 27957 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
4241biimpar 502 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
4340, 42reccld 10779 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
44 simpl 473 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
45 hvmulcl 27840 . . . . . . . . . . . 12 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4643, 44, 45syl2anc 692 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
47 norm1 28076 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
48 1le1 10640 . . . . . . . . . . . 12 1 ≤ 1
4947, 48syl6eqbr 4683 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
50 ax-his3 27911 . . . . . . . . . . . . 13 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5143, 44, 44, 50syl3anc 1324 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5213adantr 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
5352, 42rereccld 10837 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
54 hiidrcl 27922 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
5554adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ·ih 𝐴) ∈ ℝ)
5653, 55remulcld 10055 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
5751, 56eqeltrd 2699 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) ∈ ℝ)
58 normgt0 27954 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
5958biimpa 501 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
6052, 59recgt0d 10943 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
61 0re 10025 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
62 ltle 10111 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6361, 62mpan 705 . . . . . . . . . . . . . . . 16 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6453, 60, 63sylc 65 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
65 hiidge0 27925 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 ·ih 𝐴))
6753, 55, 64, 66mulge0d 10589 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
6867, 51breqtrrd 4672 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6957, 68absidd 14142 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
7040, 42recid2d 10782 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm𝐴)) = 1)
7170oveq2d 6651 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((norm𝐴) · 1))
7240, 43, 40mul12d 10230 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))))
7339sqvald 12988 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = ((norm𝐴) · (norm𝐴)))
74 normsq 27961 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
7573, 74eqtr3d 2656 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7675adantr 481 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7776oveq2d 6651 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7872, 77eqtrd 2654 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7939mulid1d 10042 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) · 1) = (norm𝐴))
8079adantr 481 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · 1) = (norm𝐴))
8171, 78, 803eqtr3rd 2663 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
8251, 69, 813eqtr4rd 2665 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
83 fveq2 6178 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (norm𝑦) = (norm‘((1 / (norm𝐴)) · 𝐴)))
8483breq1d 4654 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝑦) ≤ 1 ↔ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1))
85 oveq1 6642 . . . . . . . . . . . . . . 15 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (𝑦 ·ih 𝐴) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
8685fveq2d 6182 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
8786eqeq2d 2630 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))))
8884, 87anbi12d 746 . . . . . . . . . . . 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 3304 . . . . . . . . . . 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 1322 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9124eqeq2d 2630 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9291anbi2d 739 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9392rexbidva 3045 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9493adantr 481 . . . . . . . . . 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 247 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
96 fvex 6188 . . . . . . . . . 10 (norm𝐴) ∈ V
97 eqeq1 2624 . . . . . . . . . . . 12 (𝑥 = (norm𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
9897anbi2d 739 . . . . . . . . . . 11 (𝑥 = (norm𝐴) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9998rexbidv 3048 . . . . . . . . . 10 (𝑥 = (norm𝐴) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
10096, 99elab 3344 . . . . . . . . 9 ((norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
10195, 100sylibr 224 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
102 breq2 4648 . . . . . . . . 9 (𝑤 = (norm𝐴) → (𝑧 < 𝑤𝑧 < (norm𝐴)))
103102rspcev 3304 . . . . . . . 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 488 . . . . . . 7 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
105104adantlr 750 . . . . . 6 ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
106105ex 450 . . . . 5 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
107106ralrimiva 2963 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
108 supxr2 12129 . . . 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 1322 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (norm𝐴))
1108, 109eqtrd 2654 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = (norm𝐴))
111 nmfn0 28816 . . . 4 (normfn‘( ℋ × {0})) = 0
112 bra0 28779 . . . . 5 (bra‘0) = ( ℋ × {0})
113112fveq2i 6181 . . . 4 (normfn‘(bra‘0)) = (normfn‘( ℋ × {0}))
114 norm0 27955 . . . 4 (norm‘0) = 0
115111, 113, 1143eqtr4i 2652 . . 3 (normfn‘(bra‘0)) = (norm‘0)
116115a1i 11 . 2 (𝐴 ∈ ℋ → (normfn‘(bra‘0)) = (norm‘0))
1174, 110, 116pm2.61ne 2876 1 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  wss 3567  {csn 4168   class class class wbr 4644   × cxp 5102  wf 5872  cfv 5876  (class class class)co 6635  supcsup 8331  cc 9919  cr 9920  0cc0 9921  1c1 9922   · cmul 9926  *cxr 10058   < clt 10059  cle 10060   / cdiv 10669  2c2 11055  cexp 12843  abscabs 13955  chil 27746   · csm 27748   ·ih csp 27749  normcno 27750  0c0v 27751  normfncnmf 27778  bracbr 27783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001  ax-hilex 27826  ax-hfvadd 27827  ax-hvcom 27828  ax-hvass 27829  ax-hv0cl 27830  ax-hvaddid 27831  ax-hfvmul 27832  ax-hvmulid 27833  ax-hvmulass 27834  ax-hvdistr1 27835  ax-hvdistr2 27836  ax-hvmul0 27837  ax-hfi 27906  ax-his1 27909  ax-his2 27910  ax-his3 27911  ax-his4 27912
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-starv 15937  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-hom 15947  df-cco 15948  df-rest 16064  df-topn 16065  df-0g 16083  df-gsum 16084  df-topgen 16085  df-pt 16086  df-prds 16089  df-xrs 16143  df-qtop 16148  df-imas 16149  df-xps 16151  df-mre 16227  df-mrc 16228  df-acs 16230  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-mulg 17522  df-cntz 17731  df-cmn 18176  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-cnfld 19728  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cld 20804  df-ntr 20805  df-cls 20806  df-cn 21012  df-cnp 21013  df-t1 21099  df-haus 21100  df-tx 21346  df-hmeo 21539  df-xms 22106  df-ms 22107  df-tms 22108  df-grpo 27317  df-gid 27318  df-ginv 27319  df-gdiv 27320  df-ablo 27369  df-vc 27384  df-nv 27417  df-va 27420  df-ba 27421  df-sm 27422  df-0v 27423  df-vs 27424  df-nmcv 27425  df-ims 27426  df-dip 27526  df-ph 27638  df-hnorm 27795  df-hba 27796  df-hvsub 27798  df-nmfn 28674  df-lnfn 28677  df-bra 28679
This theorem is referenced by:  brabn  28935
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