Theorem branmfn 32714

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 6893	. . 3 ⊢ (𝐴 = 0ℎ → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0ℎ)))
2		fveq2 6888	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
3	1, 2	eqeq12d 2776	. 2 ⊢ (𝐴 = 0ℎ → ((normfn‘(bra‘𝐴)) = (normℎ‘𝐴) ↔ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)))
4		brafn 32556	. . . . 5 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
5		nmfnval 32485	. . . . 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 18	. . . 4 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
7	6	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8		nmfnsetre 32486	. . . . . . . 8 ⊢ ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
9	4, 8	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
10		ressxr 11358	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
11	9, 10	sstrdi 3921	. . . . . 6 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
12		normcl 31734	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
13	12	rexrd 11364	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ*)
14	11, 13	jca 521	. . . . 5 ⊢ (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
15	14	adantr 486	. . . 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 2764	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
18	17	anbi2d 642	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
19	18	rexbidv 3186	. . . . . . . 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 3632	. . . . . . 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 23	. . . . . . . . . . . . 13 ⊢ (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
22		braval 32553	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
23	22	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
24	23	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
25	21, 24	sylan9eqr 2817	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
26		bcs2 31791	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
27	26	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
28	27	ancom1s 666	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
29	28	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
30	25, 29	eqbrtrd 5126	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))
31	30	exp41 440	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (normℎ‘𝐴)))))
32	31	imp4a 428	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))))
33	32	rexlimdv 3161	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴)))
34	33	imp 412	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (normℎ‘𝐴))
35	20, 34	sylan2b 606	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (normℎ‘𝐴))
36	35	ralrimiva 3154	. . . . 5 ⊢ (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
38	12	recnd 11342	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ)
39	38	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
40		normne0 31739	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
41	40	biimpar 483	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
42	39, 41	reccld 12091	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
43		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
44		hvmulcl 31622	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
45	42, 43, 44	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
46		norm1 31858	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
47		1le1 11949	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
48	46, 47	eqbrtrdi 5143	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
49		ax-his3 31693	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
51	12	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
52	51, 41	rereccld 12149	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
53		hiidrcl 31704	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
54	53	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ·ih 𝐴) ∈ ℝ)
55	52, 54	remulcld 11344	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
56	50, 55	eqeltrd 2860	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) ∈ ℝ)
57		normgt0 31736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
58	57	biimpa 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
59	51, 58	recgt0d 12256	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
60		0re 11315	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
61		ltle 11403	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
62	60, 61	mpan 703	. . . . . . . . . . . . . . . 16 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
63	52, 59, 62	sylc 66	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
64		hiidge0 31707	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
65	64	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (𝐴 ·ih 𝐴))
66	52, 54, 63, 65	mulge0d 11898	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
67	66, 50	breqtrrd 5132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
68	56, 67	absidd 15597	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
69	39, 41	recid2d 12094	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1)
70	69	oveq2d 7438	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((normℎ‘𝐴) · 1))
71	39, 42, 39	mul12d 11524	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))))
72	38	sqvald 14291	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = ((normℎ‘𝐴) · (normℎ‘𝐴)))
73		normsq 31743	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
74	72, 73	eqtr3d 2797	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
75	74	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
76	75	oveq2d 7438	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
77	71, 76	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
78	38	mulridd 11331	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
79	78	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
80	70, 77, 79	3eqtr3rd 2804	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
81	50, 68, 80	3eqtr4rd 2806	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
82		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (normℎ‘𝑦) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)))
83	82	breq1d 5112	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1))
84		fvoveq1 7445	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
85	84	eqeq2d 2771	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 850	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
89	23	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
90	89	anbi2d 642	. . . . . . . . . . . 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 3184	. . . . . . . . . . 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 486	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
94		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑥 = (normℎ‘𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
95	94	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑥 = (normℎ‘𝐴) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
96	95	rexbidv 3186	. . . . . . . . 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 3634	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
98		breq2 5106	. . . . . . . . 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 592	. . . . . . 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 728	. . . . . 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 418	. . . . 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 3154	. . . 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 13449	. . . 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 850	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
106	7, 105	eqtrd 2795	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
107		nmfn0 32596	. . . 4 ⊢ (normfn‘( ℋ × {0})) = 0
108		bra0 32559	. . . . 5 ⊢ (bra‘0ℎ) = ( ℋ × {0})
109	108	fveq2i 6891	. . . 4 ⊢ (normfn‘(bra‘0ℎ)) = (normfn‘( ℋ × {0}))
110		norm0 31737	. . . 4 ⊢ (normℎ‘0ℎ) = 0
111	107, 109, 110	3eqtr4i 2793	. . 3 ⊢ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)
112	111	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ))
113	3, 106, 112	pm2.61ne 3040	1 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896  {csn 4584   class class class wbr 5102   × cxp 5648  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  supcsup 9437  ℂcc 11203  ℝcr 11204  0cc0 11205  1c1 11206   · cmul 11210  ℝ^*cxr 11347   < clt 11348   ≤ cle 11349   / cdiv 11978  2c2 12402  ↑cexp 14209  abscabs 15408   ℋchba 31528   ·_ℎ csm 31530   ·_ih csp 31531  norm_ℎcno 31532  0_ℎc0v 31533  norm_fncnmf 31560  bracbr 31565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-inf2 9647  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282  ax-pre-sup 11283  ax-addf 11284  ax-mulf 11285  ax-hilex 31608  ax-hfvadd 31609  ax-hvcom 31610  ax-hvass 31611  ax-hv0cl 31612  ax-hvaddid 31613  ax-hfvmul 31614  ax-hvmulid 31615  ax-hvmulass 31616  ax-hvdistr1 31617  ax-hvdistr2 31618  ax-hvmul0 31619  ax-hfi 31688  ax-his1 31691  ax-his2 31692  ax-his3 31693  ax-his4 31694

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-of 7695  df-om 7880  df-1st 8003  df-2nd 8004  df-supp 8182  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-er 8722  df-map 8854  df-ixp 8931  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-fsupp 9359  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9509  df-card 10025  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-div 11979  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-uz 12971  df-q 13081  df-rp 13126  df-xneg 13246  df-xadd 13247  df-xmul 13248  df-ioo 13485  df-icc 13488  df-fz 13645  df-fzo 13794  df-seq 14150  df-exp 14210  df-hash 14480  df-cj 15273  df-re 15274  df-im 15275  df-sqrt 15409  df-abs 15410  df-clim 15662  df-sum 15861  df-struct 17332  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-ress 17416  df-plusg 17448  df-mulr 17449  df-starv 17450  df-sca 17451  df-vsca 17452  df-ip 17453  df-tset 17454  df-ple 17455  df-ds 17457  df-unif 17458  df-hom 17459  df-cco 17460  df-rest 17600  df-topn 17601  df-0g 17619  df-gsum 17620  df-topgen 17621  df-pt 17622  df-prds 17625  df-xrs 17681  df-qtop 17686  df-imas 17687  df-xps 17689  df-mre 17763  df-mrc 17764  df-acs 17766  df-mgm 18823  df-sgrp 18915  df-mnd 18931  df-submnd 18986  df-mulg 19285  df-cntz 19538  df-cmn 20003  df-psmet 21677  df-xmet 21678  df-met 21679  df-bl 21680  df-mopn 21681  df-cnfld 21686  df-top 23219  df-topon 23236  df-topsp 23258  df-bases 23271  df-cld 23344  df-ntr 23345  df-cls 23346  df-cn 23552  df-cnp 23553  df-t1 23639  df-haus 23640  df-tx 23888  df-hmeo 24081  df-xms 24646  df-ms 24647  df-tms 24648  df-grpo 31102  df-gid 31103  df-ginv 31104  df-gdiv 31105  df-ablo 31154  df-vc 31168  df-nv 31201  df-va 31204  df-ba 31205  df-sm 31206  df-0v 31207  df-vs 31208  df-nmcv 31209  df-ims 31210  df-dip 31310  df-ph 31422  df-hnorm 31577  df-hba 31578  df-hvsub 31580  df-nmfn 32454  df-lnfn 32457  df-bra 32459

This theorem is used by:  brabn  32715