Theorem branmfn 28348

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		fveq2 6103	. . . 4 ⊢ (𝐴 = 0ℎ → (bra‘𝐴) = (bra‘0ℎ))
2	1	fveq2d 6107	. . 3 ⊢ (𝐴 = 0ℎ → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0ℎ)))
3		fveq2 6103	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
4	2, 3	eqeq12d 2625	. 2 ⊢ (𝐴 = 0ℎ → ((normfn‘(bra‘𝐴)) = (normℎ‘𝐴) ↔ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)))
5		brafn 28190	. . . . 5 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
6		nmfnval 28119	. . . . 5 ⊢ ((bra‘𝐴): ℋ⟶ℂ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
7	5, 6	syl 17	. . . 4 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
8	7	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
9		nmfnsetre 28120	. . . . . . . 8 ⊢ ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
10	5, 9	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
11		ressxr 9962	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
12	10, 11	syl6ss 3580	. . . . . 6 ⊢ (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
13		normcl 27366	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
14	13	rexrd 9968	. . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ*)
15	12, 14	jca 553	. . . . 5 ⊢ (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
16	15	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘𝐴) ∈ ℝ*))
17		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
18		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
19	18	anbi2d 736	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
20	19	rexbidv 3034	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
21	17, 20	elab 3319	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
22		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
23		braval 28187	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
24	23	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
26	22, 25	sylan9eqr 2666	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
27		bcs2 27423	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
28	27	3expa 1257	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
29	28	ancom1s 843	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
30	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (normℎ‘𝐴))
31	26, 30	eqbrtrd 4605	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))
32	31	exp41 636	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (normℎ‘𝐴)))))
33	32	imp4a 612	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴))))
34	33	rexlimdv 3012	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (normℎ‘𝐴)))
35	34	imp 444	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (normℎ‘𝐴))
36	21, 35	sylan2b 491	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (normℎ‘𝐴))
37	36	ralrimiva 2949	. . . . 5 ⊢ (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
38	37	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (normℎ‘𝐴))
39	13	recnd 9947	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ)
40	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
41		normne0 27371	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
42	41	biimpar 501	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
43	40, 42	reccld 10673	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
44		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
45		hvmulcl 27254	. . . . . . . . . . . 12 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
46	43, 44, 45	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
47		norm1 27490	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
48		1le1 10534	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
49	47, 48	syl6eqbr 4622	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
50		ax-his3 27325	. . . . . . . . . . . . 13 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
51	43, 44, 44, 50	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
52	13	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
53	52, 42	rereccld 10731	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
54		hiidrcl 27336	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
55	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝐴 ·ih 𝐴) ∈ ℝ)
56	53, 55	remulcld 9949	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
57	51, 56	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴) ∈ ℝ)
58		normgt0 27368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
59	58	biimpa 500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
60	52, 59	recgt0d 10837	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
61		0re 9919	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
62		ltle 10005	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
63	61, 62	mpan 702	. . . . . . . . . . . . . . . 16 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
64	53, 60, 63	sylc 63	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
65		hiidge0 27339	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (𝐴 ·ih 𝐴))
67	53, 55, 64, 66	mulge0d 10483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
68	67, 51	breqtrrd 4611	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
69	57, 68	absidd 14009	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
70	40, 42	recid2d 10676	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1)
71	70	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((normℎ‘𝐴) · 1))
72	40, 43, 40	mul12d 10124	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))))
73	39	sqvald 12867	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = ((normℎ‘𝐴) · (normℎ‘𝐴)))
74		normsq 27375	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
75	73, 74	eqtr3d 2646	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
76	75	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · (normℎ‘𝐴)) = (𝐴 ·ih 𝐴))
77	76	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · ((normℎ‘𝐴) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
78	72, 77	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
79	39	mulid1d 9936	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
80	79	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴) · 1) = (normℎ‘𝐴))
81	71, 78, 80	3eqtr3rd 2653	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = ((1 / (normℎ‘𝐴)) · (𝐴 ·ih 𝐴)))
82	51, 69, 81	3eqtr4rd 2655	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
83		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (normℎ‘𝑦) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)))
84	83	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1))
85		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (𝑦 ·ih 𝐴) = (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))
86	85	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴)))
87	86	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) → ((normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (normℎ‘𝐴) = (abs‘(((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ·ih 𝐴))))
88	84, 87	anbi12d 743	. . . . . . . . . . . 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)))))
89	88	rspcev 3282	. . . . . . . . . . 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))))
90	46, 49, 82, 89	syl12anc 1316	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
91	24	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴))))
92	91	anbi2d 736	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
93	92	rexbidva 3031	. . . . . . . . . . 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)))))
94	93	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)))))
95	90, 94	mpbird 246	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
96		fvex 6113	. . . . . . . . . 10 ⊢ (normℎ‘𝐴) ∈ V
97		eqeq1 2614	. . . . . . . . . . . 12 ⊢ (𝑥 = (normℎ‘𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
98	97	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑥 = (normℎ‘𝐴) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
99	98	rexbidv 3034	. . . . . . . . . 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)))))
100	96, 99	elab 3319	. . . . . . . . 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))))
101	95, 100	sylibr 223	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
102		breq2 4587	. . . . . . . . 9 ⊢ (𝑤 = (normℎ‘𝐴) → (𝑧 < 𝑤 ↔ 𝑧 < (normℎ‘𝐴)))
103	102	rspcev 3282	. . . . . . . 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)
104	101, 103	sylan 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
105	104	adantlr 747	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (normℎ‘𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
106	105	ex 449	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ 𝑧 ∈ ℝ) → (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
107	106	ralrimiva 2949	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ∀𝑧 ∈ ℝ (𝑧 < (normℎ‘𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
108		supxr2 12016	. . . 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))
109	16, 38, 107, 108	syl12anc 1316	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (normℎ‘𝐴))
110	8, 109	eqtrd 2644	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
111		nmfn0 28230	. . . 4 ⊢ (normfn‘( ℋ × {0})) = 0
112		bra0 28193	. . . . 5 ⊢ (bra‘0ℎ) = ( ℋ × {0})
113	112	fveq2i 6106	. . . 4 ⊢ (normfn‘(bra‘0ℎ)) = (normfn‘( ℋ × {0}))
114		norm0 27369	. . . 4 ⊢ (normℎ‘0ℎ) = 0
115	111, 113, 114	3eqtr4i 2642	. . 3 ⊢ (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ)
116	115	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘0ℎ)) = (normℎ‘0ℎ))
117	4, 110, 116	pm2.61ne 2867	1 ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   / cdiv 10563  2c2 10947  ↑cexp 12722  abscabs 13822   ℋchil 27160   ·_ℎ csm 27162   ·_ih csp 27163  norm_ℎcno 27164  0_ℎc0v 27165  norm_fncnmf 27192  bracbr 27197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895  ax-hilex 27240  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmulass 27248  ax-hvdistr1 27249  ax-hvdistr2 27250  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his2 27324  ax-his3 27325  ax-his4 27326

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840  df-dip 26940  df-ph 27052  df-hnorm 27209  df-hba 27210  df-hvsub 27212  df-nmfn 28088  df-lnfn 28091  df-bra 28093

This theorem is referenced by:  brabn  28349