Theorem branmfn 30821

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)) = (normℎ‘𝐴))

Proof of Theorem branmfn

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6839	. . 3 ⊢ (𝐴 = 0ℎ → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0ℎ)))
2		fveq2 6834	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
3	1, 2	eqeq12d 2753	. 2 ⊢ (𝐴 = 0ℎ → ((normfn‘(bra‘𝐴)) = (normℎ‘𝐴) ↔ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)))
4		brafn 30663	. . . . 5 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5		nmfnval 30592	. . . . 5 ⊢ ((bra‘𝐴): ℋ⟶ℂ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
6	4, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
7	6	adantr 482	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8		nmfnsetre 30593	. . . . . . . 8 ⊢ ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
9	4, 8	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
10		ressxr 11129	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
11	9, 10	sstrdi 3951	. . . . . 6 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12		normcl 29841	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
13	12	rexrd 11135	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ*)
14	11, 13	jca 513	. . . . 5 ⊢ (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
15	14	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
16		vex 3447	. . . . . . . 8 ⊢ 𝑧 ∈ V
17		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
18	17	anbi2d 630	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
19	18	rexbidv 3173	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
20	16, 19	elab 3625	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
21		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
22		braval 30660	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
23	22	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
24	23	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
25	21, 24	sylan9eqr 2799	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26		bcs2 29898	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
27	26	3expa 1118	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
28	27	ancom1s 651	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
29	28	adantr 482	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
30	25, 29	eqbrtrd 5122	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))
31	30	exp41 436	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (normℎ‘𝐴)))))
32	31	imp4a 424	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))))
33	32	rexlimdv 3148	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴)))
34	33	imp 408	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (normℎ‘𝐴))
35	20, 34	sylan2b 595	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (normℎ‘𝐴))
36	35	ralrimiva 3141	. . . . 5 ⊢ (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
37	36	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
38	12	recnd 11113	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ)
39	38	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
40		normne0 29846	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
41	40	biimpar 479	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
42	39, 41	reccld 11854	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
43		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
44		hvmulcl 29729	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
45	42, 43, 44	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
46		norm1 29965	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
47		1le1 11713	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
48	46, 47	eqbrtrdi 5139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
49		ax-his3 29800	. . . . . . . . . . . . 13 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
50	42, 43, 43, 49	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
51	12	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
52	51, 41	rereccld 11912	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
53		hiidrcl 29811	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
54	53	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ·ih 𝐴) ∈ ℝ)
55	52, 54	remulcld 11115	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
56	50, 55	eqeltrd 2838	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) ∈ ℝ)
57		normgt0 29843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
58	57	biimpa 478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
59	51, 58	recgt0d 12019	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
60		0re 11087	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
61		ltle 11173	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
62	60, 61	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
63	52, 59, 62	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
64		hiidge0 29814	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
65	64	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (𝐴 ·ih 𝐴))
66	52, 54, 63, 65	mulge0d 11662	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
67	66, 50	breqtrrd 5128	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
68	56, 67	absidd 15238	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
69	39, 41	recid2d 11857	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1)
70	69	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((normℎ‘𝐴) · 1))
71	39, 42, 39	mul12d 11294	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))))
72	38	sqvald 13971	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = ((normℎ‘𝐴) · (normℎ‘𝐴)))
73		normsq 29850	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
74	72, 73	eqtr3d 2779	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
75	74	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
76	75	oveq2d 7362	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
77	71, 76	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
78	38	mulid1d 11102	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
79	78	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
80	70, 77, 79	3eqtr3rd 2786	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
81	50, 68, 80	3eqtr4rd 2788	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
82		fveq2 6834	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (normℎ‘𝑦) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)))
83	82	breq1d 5110	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1))
84		fvoveq1 7369	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
85	84	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))))
86	83, 85	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))) ↔ ((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))))
87	86	rspcev 3576	. . . . . . . . . . 11 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ ((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
88	45, 48, 81, 87	syl12anc 835	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
89	23	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
90	89	anbi2d 630	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
91	90	rexbidva 3171	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
92	91	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
93	88, 92	mpbird 257	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
94		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑥 = (normℎ‘𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
95	94	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑥 = (normℎ‘𝐴) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
96	95	rexbidv 3173	. . . . . . . . 9 ⊢ (𝑥 = (normℎ‘𝐴) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
97	39, 93, 96	elabd 3628	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
98		breq2 5104	. . . . . . . . 9 ⊢ (𝑤 = (normℎ‘𝐴) → (𝑧 < 𝑤 ↔ 𝑧 < (normℎ‘𝐴)))
99	98	rspcev 3576	. . . . . . . 8 ⊢ (((normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
100	97, 99	sylan 581	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
101	100	adantlr 713	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
102	101	ex 414	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) → (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
103	102	ralrimiva 3141	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ ℝ (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
104		supxr2 13158	. . . 4 ⊢ ((({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*) ∧ (∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴) ∧ ∀𝑧 ∈ ℝ (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
105	15, 37, 103, 104	syl12anc 835	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
106	7, 105	eqtrd 2777	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
107		nmfn0 30703	. . . 4 ⊢ (normfn‘( ℋ × {0})) = 0
108		bra0 30666	. . . . 5 ⊢ (bra‘0ℎ) = ( ℋ × {0})
109	108	fveq2i 6837	. . . 4 ⊢ (normfn‘(bra‘0ℎ)) = (normfn‘( ℋ × {0}))
110		norm0 29844	. . . 4 ⊢ (normℎ‘0ℎ) = 0
111	107, 109, 110	3eqtr4i 2775	. . 3 ⊢ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)
112	111	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ))
113	3, 106, 112	pm2.61ne 3028	1 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541   ∈ wcel 2106  {cab 2714   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ⊆ wss 3905  {csn 4581   class class class wbr 5100   × cxp 5625  ⟶wf 6484  ‘cfv 6488  (class class class)co 7346  supcsup 9306  ℂcc 10979  ℝcr 10980  0cc0 10981  1c1 10982   · cmul 10986  ℝ^*cxr 11118   < clt 11119   ≤ cle 11120   / cdiv 11742  2c2 12138  ↑cexp 13892  abscabs 15049   ℋchba 29635   ·_ℎ csm 29637   ·_ih csp 29638  norm_ℎcno 29639  0_ℎc0v 29640  norm_fncnmf 29667  bracbr 29672

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 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659  ax-inf2 9507  ax-cnex 11037  ax-resscn 11038  ax-1cn 11039  ax-icn 11040  ax-addcl 11041  ax-addrcl 11042  ax-mulcl 11043  ax-mulrcl 11044  ax-mulcom 11045  ax-addass 11046  ax-mulass 11047  ax-distr 11048  ax-i2m1 11049  ax-1ne0 11050  ax-1rid 11051  ax-rnegex 11052  ax-rrecex 11053  ax-cnre 11054  ax-pre-lttri 11055  ax-pre-lttrn 11056  ax-pre-ltadd 11057  ax-pre-mulgt0 11058  ax-pre-sup 11059  ax-addf 11060  ax-mulf 11061  ax-hilex 29715  ax-hfvadd 29716  ax-hvcom 29717  ax-hvass 29718  ax-hv0cl 29719  ax-hvaddid 29720  ax-hfvmul 29721  ax-hvmulid 29722  ax-hvmulass 29723  ax-hvdistr1 29724  ax-hvdistr2 29725  ax-hvmul0 29726  ax-hfi 29795  ax-his1 29798  ax-his2 29799  ax-his3 29800  ax-his4 29801

This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4861  df-int 4903  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5184  df-tr 5218  df-id 5525  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-se 5583  df-we 5584  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-pred 6246  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7302  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7604  df-om 7790  df-1st 7908  df-2nd 7909  df-supp 8057  df-frecs 8176  df-wrecs 8207  df-recs 8281  df-rdg 8320  df-1o 8376  df-2o 8377  df-er 8578  df-map 8697  df-ixp 8766  df-en 8814  df-dom 8815  df-sdom 8816  df-fin 8817  df-fsupp 9236  df-fi 9277  df-sup 9308  df-inf 9309  df-oi 9376  df-card 9805  df-pnf 11121  df-mnf 11122  df-xr 11123  df-ltxr 11124  df-le 11125  df-sub 11317  df-neg 11318  df-div 11743  df-nn 12084  df-2 12146  df-3 12147  df-4 12148  df-5 12149  df-6 12150  df-7 12151  df-8 12152  df-9 12153  df-n0 12344  df-z 12430  df-dec 12548  df-uz 12693  df-q 12799  df-rp 12841  df-xneg 12958  df-xadd 12959  df-xmul 12960  df-ioo 13193  df-icc 13196  df-fz 13350  df-fzo 13493  df-seq 13832  df-exp 13893  df-hash 14155  df-cj 14914  df-re 14915  df-im 14916  df-sqrt 15050  df-abs 15051  df-clim 15301  df-sum 15502  df-struct 16950  df-sets 16967  df-slot 16985  df-ndx 16997  df-base 17015  df-ress 17044  df-plusg 17077  df-mulr 17078  df-starv 17079  df-sca 17080  df-vsca 17081  df-ip 17082  df-tset 17083  df-ple 17084  df-ds 17086  df-unif 17087  df-hom 17088  df-cco 17089  df-rest 17235  df-topn 17236  df-0g 17254  df-gsum 17255  df-topgen 17256  df-pt 17257  df-prds 17260  df-xrs 17315  df-qtop 17320  df-imas 17321  df-xps 17323  df-mre 17397  df-mrc 17398  df-acs 17400  df-mgm 18428  df-sgrp 18477  df-mnd 18488  df-submnd 18533  df-mulg 18802  df-cntz 19024  df-cmn 19488  df-psmet 20699  df-xmet 20700  df-met 20701  df-bl 20702  df-mopn 20703  df-cnfld 20708  df-top 22153  df-topon 22170  df-topsp 22192  df-bases 22206  df-cld 22280  df-ntr 22281  df-cls 22282  df-cn 22488  df-cnp 22489  df-t1 22575  df-haus 22576  df-tx 22823  df-hmeo 23016  df-xms 23583  df-ms 23584  df-tms 23585  df-grpo 29209  df-gid 29210  df-ginv 29211  df-gdiv 29212  df-ablo 29261  df-vc 29275  df-nv 29308  df-va 29311  df-ba 29312  df-sm 29313  df-0v 29314  df-vs 29315  df-nmcv 29316  df-ims 29317  df-dip 29417  df-ph 29529  df-hnorm 29684  df-hba 29685  df-hvsub 29687  df-nmfn 30561  df-lnfn 30564  df-bra 30566

This theorem is referenced by:  brabn  30822