Theorem branmfn 29679

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 6502	. . 3 ⊢ (𝐴 = 0ℎ → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0ℎ)))
2		fveq2 6497	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
3	1, 2	eqeq12d 2788	. 2 ⊢ (𝐴 = 0ℎ → ((normfn‘(bra‘𝐴)) = (normℎ‘𝐴) ↔ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)))
4		brafn 29521	. . . . 5 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5		nmfnval 29450	. . . . 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 473	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8		nmfnsetre 29451	. . . . . . . 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 10483	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
11	9, 10	syl6ss 3865	. . . . . 6 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12		normcl 28697	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
13	12	rexrd 10489	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ*)
14	11, 13	jca 504	. . . . 5 ⊢ (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
15	14	adantr 473	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
16		vex 3413	. . . . . . . 8 ⊢ 𝑧 ∈ V
17		eqeq1 2777	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
18	17	anbi2d 620	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
19	18	rexbidv 3237	. . . . . . . 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 3577	. . . . . . 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 29518	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
23	22	fveq2d 6501	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
24	23	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
25	21, 24	sylan9eqr 2831	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26		bcs2 28754	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
27	26	3expa 1099	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
28	27	ancom1s 641	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
29	28	adantr 473	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
30	25, 29	eqbrtrd 4948	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))
31	30	exp41 427	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (normℎ‘𝐴)))))
32	31	imp4a 415	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))))
33	32	rexlimdv 3223	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴)))
34	33	imp 398	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (normℎ‘𝐴))
35	20, 34	sylan2b 585	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (normℎ‘𝐴))
36	35	ralrimiva 3127	. . . . 5 ⊢ (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
37	36	adantr 473	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
38	12	recnd 10467	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ)
39	38	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
40		normne0 28702	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
41	40	biimpar 470	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
42	39, 41	reccld 11209	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
43		simpl 475	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
44		hvmulcl 28585	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
45	42, 43, 44	syl2anc 576	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
46		norm1 28821	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
47		1le1 11068	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
48	46, 47	syl6eqbr 4965	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
49		ax-his3 28656	. . . . . . . . . . . . 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 1352	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
51	12	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
52	51, 41	rereccld 11267	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
53		hiidrcl 28667	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
54	53	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ·ih 𝐴) ∈ ℝ)
55	52, 54	remulcld 10469	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
56	50, 55	eqeltrd 2861	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) ∈ ℝ)
57		normgt0 28699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
58	57	biimpa 469	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
59	51, 58	recgt0d 11374	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
60		0re 10440	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
61		ltle 10528	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
62	60, 61	mpan 678	. . . . . . . . . . . . . . . 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 28670	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
65	64	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (𝐴 ·ih 𝐴))
66	52, 54, 63, 65	mulge0d 11017	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
67	66, 50	breqtrrd 4954	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
68	56, 67	absidd 14642	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
69	39, 41	recid2d 11212	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1)
70	69	oveq2d 6991	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((normℎ‘𝐴) · 1))
71	39, 42, 39	mul12d 10648	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))))
72	38	sqvald 13321	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = ((normℎ‘𝐴) · (normℎ‘𝐴)))
73		normsq 28706	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
74	72, 73	eqtr3d 2811	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
75	74	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
76	75	oveq2d 6991	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
77	71, 76	eqtrd 2809	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
78	38	mulid1d 10456	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
79	78	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
80	70, 77, 79	3eqtr3rd 2818	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
81	50, 68, 80	3eqtr4rd 2820	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
82		fveq2 6497	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (normℎ‘𝑦) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)))
83	82	breq1d 4936	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1))
84		fvoveq1 6998	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
85	84	eqeq2d 2783	. . . . . . . . . . . . 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 622	. . . . . . . . . . . 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 3530	. . . . . . . . . . 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 825	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
89	23	eqeq2d 2783	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
90	89	anbi2d 620	. . . . . . . . . . . 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 3236	. . . . . . . . . . 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 473	. . . . . . . . . 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 249	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
94		fvex 6510	. . . . . . . . . 10 ⊢ (normℎ‘𝐴) ∈ V
95		eqeq1 2777	. . . . . . . . . . . 12 ⊢ (𝑥 = (normℎ‘𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
96	95	anbi2d 620	. . . . . . . . . . 11 ⊢ (𝑥 = (normℎ‘𝐴) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
97	96	rexbidv 3237	. . . . . . . . . 10 ⊢ (𝑥 = (normℎ‘𝐴) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
98	94, 97	elab 3577	. . . . . . . . 9 ⊢ ((normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
99	93, 98	sylibr 226	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
100		breq2 4930	. . . . . . . . 9 ⊢ (𝑤 = (normℎ‘𝐴) → (𝑧 < 𝑤 ↔ 𝑧 < (normℎ‘𝐴)))
101	100	rspcev 3530	. . . . . . . 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)
102	99, 101	sylan 572	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
103	102	adantlr 703	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
104	103	ex 405	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) → (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
105	104	ralrimiva 3127	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ ℝ (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
106		supxr2 12522	. . . 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))
107	15, 37, 105, 106	syl12anc 825	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
108	7, 107	eqtrd 2809	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
109		nmfn0 29561	. . . 4 ⊢ (normfn‘( ℋ × {0})) = 0
110		bra0 29524	. . . . 5 ⊢ (bra‘0ℎ) = ( ℋ × {0})
111	110	fveq2i 6500	. . . 4 ⊢ (normfn‘(bra‘0ℎ)) = (normfn‘( ℋ × {0}))
112		norm0 28700	. . . 4 ⊢ (normℎ‘0ℎ) = 0
113	109, 111, 112	3eqtr4i 2807	. . 3 ⊢ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)
114	113	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ))
115	3, 108, 114	pm2.61ne 3048	1 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2753   ≠ wne 2962  ∀wral 3083  ∃wrex 3084   ⊆ wss 3824  {csn 4436   class class class wbr 4926   × cxp 5402  ⟶wf 6182  ‘cfv 6186  (class class class)co 6975  supcsup 8698  ℂcc 10332  ℝcr 10333  0cc0 10334  1c1 10335   · cmul 10339  ℝ^*cxr 10472   < clt 10473   ≤ cle 10474   / cdiv 11097  2c2 11494  ↑cexp 13243  abscabs 14453   ℋchba 28491   ·_ℎ csm 28493   ·_ih csp 28494  norm_ℎcno 28495  0_ℎc0v 28496  norm_fncnmf 28523  bracbr 28528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412  ax-addf 10413  ax-mulf 10414  ax-hilex 28571  ax-hfvadd 28572  ax-hvcom 28573  ax-hvass 28574  ax-hv0cl 28575  ax-hvaddid 28576  ax-hfvmul 28577  ax-hvmulid 28578  ax-hvmulass 28579  ax-hvdistr1 28580  ax-hvdistr2 28581  ax-hvmul0 28582  ax-hfi 28651  ax-his1 28654  ax-his2 28655  ax-his3 28656  ax-his4 28657

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-iin 4792  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-om 7396  df-1st 7500  df-2nd 7501  df-supp 7633  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-ixp 8259  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-fsupp 8628  df-fi 8669  df-sup 8700  df-inf 8701  df-oi 8768  df-card 9161  df-cda 9387  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-4 11504  df-5 11505  df-6 11506  df-7 11507  df-8 11508  df-9 11509  df-n0 11707  df-z 11793  df-dec 11911  df-uz 12058  df-q 12162  df-rp 12204  df-xneg 12323  df-xadd 12324  df-xmul 12325  df-ioo 12557  df-icc 12560  df-fz 12708  df-fzo 12849  df-seq 13184  df-exp 13244  df-hash 13505  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-clim 14705  df-sum 14903  df-struct 16340  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-starv 16435  df-sca 16436  df-vsca 16437  df-ip 16438  df-tset 16439  df-ple 16440  df-ds 16442  df-unif 16443  df-hom 16444  df-cco 16445  df-rest 16551  df-topn 16552  df-0g 16570  df-gsum 16571  df-topgen 16572  df-pt 16573  df-prds 16576  df-xrs 16630  df-qtop 16635  df-imas 16636  df-xps 16638  df-mre 16728  df-mrc 16729  df-acs 16731  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-submnd 17817  df-mulg 18025  df-cntz 18231  df-cmn 18681  df-psmet 20255  df-xmet 20256  df-met 20257  df-bl 20258  df-mopn 20259  df-cnfld 20264  df-top 21222  df-topon 21239  df-topsp 21261  df-bases 21274  df-cld 21347  df-ntr 21348  df-cls 21349  df-cn 21555  df-cnp 21556  df-t1 21642  df-haus 21643  df-tx 21890  df-hmeo 22083  df-xms 22649  df-ms 22650  df-tms 22651  df-grpo 28063  df-gid 28064  df-ginv 28065  df-gdiv 28066  df-ablo 28115  df-vc 28129  df-nv 28162  df-va 28165  df-ba 28166  df-sm 28167  df-0v 28168  df-vs 28169  df-nmcv 28170  df-ims 28171  df-dip 28271  df-ph 28383  df-hnorm 28540  df-hba 28541  df-hvsub 28543  df-nmfn 29419  df-lnfn 29422  df-bra 29424

This theorem is referenced by:  brabn  29680