Theorem branmfn 32041

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 6866	. . 3 ⊢ (𝐴 = 0ℎ → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0ℎ)))
2		fveq2 6861	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
3	1, 2	eqeq12d 2746	. 2 ⊢ (𝐴 = 0ℎ → ((normfn‘(bra‘𝐴)) = (normℎ‘𝐴) ↔ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)))
4		brafn 31883	. . . . 5 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5		nmfnval 31812	. . . . 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 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8		nmfnsetre 31813	. . . . . . . 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 11225	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
11	9, 10	sstrdi 3962	. . . . . 6 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12		normcl 31061	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
13	12	rexrd 11231	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ*)
14	11, 13	jca 511	. . . . 5 ⊢ (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
15	14	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
16		vex 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
17		eqeq1 2734	. . . . . . . . . 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 3158	. . . . . . . 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 3649	. . . . . . 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 31880	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
23	22	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
24	23	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
25	21, 24	sylan9eqr 2787	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26		bcs2 31118	. . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
30	25, 29	eqbrtrd 5132	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))
31	30	exp41 434	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (normℎ‘𝐴)))))
32	31	imp4a 422	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))))
33	32	rexlimdv 3133	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴)))
34	33	imp 406	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (normℎ‘𝐴))
35	20, 34	sylan2b 594	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (normℎ‘𝐴))
36	35	ralrimiva 3126	. . . . 5 ⊢ (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
38	12	recnd 11209	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ)
39	38	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
40		normne0 31066	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
41	40	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
42	39, 41	reccld 11958	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
43		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
44		hvmulcl 30949	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
45	42, 43, 44	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
46		norm1 31185	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
47		1le1 11813	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
48	46, 47	eqbrtrdi 5149	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
49		ax-his3 31020	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
51	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
52	51, 41	rereccld 12016	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
53		hiidrcl 31031	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
54	53	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ·ih 𝐴) ∈ ℝ)
55	52, 54	remulcld 11211	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
56	50, 55	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) ∈ ℝ)
57		normgt0 31063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
58	57	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
59	51, 58	recgt0d 12124	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
60		0re 11183	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
61		ltle 11269	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
62	60, 61	mpan 690	. . . . . . . . . . . . . . . 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 31034	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
65	64	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (𝐴 ·ih 𝐴))
66	52, 54, 63, 65	mulge0d 11762	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
67	66, 50	breqtrrd 5138	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
68	56, 67	absidd 15396	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
69	39, 41	recid2d 11961	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1)
70	69	oveq2d 7406	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((normℎ‘𝐴) · 1))
71	39, 42, 39	mul12d 11390	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))))
72	38	sqvald 14115	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = ((normℎ‘𝐴) · (normℎ‘𝐴)))
73		normsq 31070	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
74	72, 73	eqtr3d 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
75	74	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
76	75	oveq2d 7406	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
77	71, 76	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
78	38	mulridd 11198	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
79	78	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
80	70, 77, 79	3eqtr3rd 2774	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
81	50, 68, 80	3eqtr4rd 2776	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
82		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (normℎ‘𝑦) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)))
83	82	breq1d 5120	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1))
84		fvoveq1 7413	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
85	84	eqeq2d 2741	. . . . . . . . . . . . 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 3591	. . . . . . . . . . 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 836	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
89	23	eqeq2d 2741	. . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 480	. . . . . . . . . 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 2734	. . . . . . . . . . 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 3158	. . . . . . . . 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 3651	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
98		breq2 5114	. . . . . . . . 9 ⊢ (𝑤 = (normℎ‘𝐴) → (𝑧 < 𝑤 ↔ 𝑧 < (normℎ‘𝐴)))
99	98	rspcev 3591	. . . . . . . 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 580	. . . . . . 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 715	. . . . . 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 412	. . . . 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 3126	. . . 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 13281	. . . 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 836	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
106	7, 105	eqtrd 2765	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
107		nmfn0 31923	. . . 4 ⊢ (normfn‘( ℋ × {0})) = 0
108		bra0 31886	. . . . 5 ⊢ (bra‘0ℎ) = ( ℋ × {0})
109	108	fveq2i 6864	. . . 4 ⊢ (normfn‘(bra‘0ℎ)) = (normfn‘( ℋ × {0}))
110		norm0 31064	. . . 4 ⊢ (normℎ‘0ℎ) = 0
111	107, 109, 110	3eqtr4i 2763	. . 3 ⊢ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)
112	111	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ))
113	3, 106, 112	pm2.61ne 3011	1 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  {csn 4592   class class class wbr 5110   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  supcsup 9398  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   / cdiv 11842  2c2 12248  ↑cexp 14033  abscabs 15207   ℋchba 30855   ·_ℎ csm 30857   ·_ih csp 30858  norm_ℎcno 30859  0_ℎc0v 30860  norm_fncnmf 30887  bracbr 30892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154  ax-mulf 11155  ax-hilex 30935  ax-hfvadd 30936  ax-hvcom 30937  ax-hvass 30938  ax-hv0cl 30939  ax-hvaddid 30940  ax-hfvmul 30941  ax-hvmulid 30942  ax-hvmulass 30943  ax-hvdistr1 30944  ax-hvdistr2 30945  ax-hvmul0 30946  ax-hfi 31015  ax-his1 31018  ax-his2 31019  ax-his3 31020  ax-his4 31021

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-icc 13320  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-rest 17392  df-topn 17393  df-0g 17411  df-gsum 17412  df-topgen 17413  df-pt 17414  df-prds 17417  df-xrs 17472  df-qtop 17477  df-imas 17478  df-xps 17480  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-mulg 19007  df-cntz 19256  df-cmn 19719  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cld 22913  df-ntr 22914  df-cls 22915  df-cn 23121  df-cnp 23122  df-t1 23208  df-haus 23209  df-tx 23456  df-hmeo 23649  df-xms 24215  df-ms 24216  df-tms 24217  df-grpo 30429  df-gid 30430  df-ginv 30431  df-gdiv 30432  df-ablo 30481  df-vc 30495  df-nv 30528  df-va 30531  df-ba 30532  df-sm 30533  df-0v 30534  df-vs 30535  df-nmcv 30536  df-ims 30537  df-dip 30637  df-ph 30749  df-hnorm 30904  df-hba 30905  df-hvsub 30907  df-nmfn 31781  df-lnfn 31784  df-bra 31786

This theorem is referenced by:  brabn  32042