tcphcph - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tcphcph		Structured version Visualization version GIF version

Theorem tcphcph 23937

Description: The standard definition of a norm turns any pre-Hilbert space over a subfield of ℂ_fld closed under square roots of nonnegative reals into a subcomplex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)

**Hypotheses**
Ref	Expression
tcphval.n	⊢ 𝐺 = (toℂPreHil‘𝑊)
tcphcph.v	⊢ 𝑉 = (Base‘𝑊)
tcphcph.f	⊢ 𝐹 = (Scalar‘𝑊)
tcphcph.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
tcphcph.2	⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
tcphcph.h	⊢ , = (·_𝑖‘𝑊)
tcphcph.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)
tcphcph.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))

**Assertion**
Ref	Expression
tcphcph	⊢ (𝜑 → 𝐺 ∈ ℂPreHil)

Distinct variable groups: 𝑥, , 𝑥,𝐹 𝑥,𝐺 𝑥,𝑉 𝜑,𝑥 𝑥,𝑊 Allowed substitution hint: 𝐾(𝑥)

Proof of Theorem tcphcph Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tcphcph.1	. . . 4 ⊢ (𝜑 → 𝑊 ∈ PreHil)
2		tcphval.n	. . . . 5 ⊢ 𝐺 = (toℂPreHil‘𝑊)
3	2	tcphphl 23927	. . . 4 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)
4	1, 3	sylib 221	. . 3 ⊢ (𝜑 → 𝐺 ∈ PreHil)
5		tcphcph.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
6		tcphcph.h	. . . . . . 7 ⊢ , = (·_𝑖‘𝑊)
7	2, 5, 6	tcphval 23918	. . . . . 6 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
8		eqid 2758	. . . . . 6 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
9		eqid 2758	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
10		phllmod 20395	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
11	1, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
12		lmodgrp 19709	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
14		tcphcph.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
15		tcphcph.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
16	2, 5, 14, 1, 15, 6	tcphcphlem3 23933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ)
17		tcphcph.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
18	16, 17	resqrtcld 14825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ ℝ)
19	18	fmpttd 6870	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶ℝ)
20		oveq12 7159	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 , 𝑥) = (𝑦 , 𝑦))
21	20	anidms 570	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 , 𝑥) = (𝑦 , 𝑦))
22	21	fveq2d 6662	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (√‘(𝑥 , 𝑥)) = (√‘(𝑦 , 𝑦)))
23		eqid 2758	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
24		fvex 6671	. . . . . . . . . 10 ⊢ (√‘(𝑥 , 𝑥)) ∈ V
25	22, 23, 24	fvmpt3i 6764	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
26	25	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
27	26	eqeq1d 2760	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
28		eqid 2758	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐹) = (Base‘𝐹)
29		phllvec 20394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
30	1, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ LVec)
31	14	lvecdrng 19945	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
32	30, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ DivRing)
33	28, 15, 32	cphsubrglem 23878	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 = (ℂ_fld ↾_s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)))
34	33	simp2d 1140	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (𝐾 ∩ ℂ))
35		inss2 4134	. . . . . . . . . . . . 13 ⊢ (𝐾 ∩ ℂ) ⊆ ℂ
36	34, 35	eqsstrdi 3946	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐹) ⊆ ℂ)
37	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (Base‘𝐹) ⊆ ℂ)
38	14, 6, 5, 28	ipcl 20398	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
39	38	3anidm23 1418	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
40	1, 39	sylan 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
41	37, 40	sseldd 3893	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ ℂ)
42	41	sqrtcld 14845	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (√‘(𝑦 , 𝑦)) ∈ ℂ)
43		sqeq0 13536	. . . . . . . . 9 ⊢ ((√‘(𝑦 , 𝑦)) ∈ ℂ → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
45	41	sqsqrtd 14847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦))↑2) = (𝑦 , 𝑦))
46	2, 5, 14, 1, 15	phclm 23932	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
47	14	clm0 23773	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂMod → 0 = (0_g‘𝐹))
48	46, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (0_g‘𝐹))
49	48	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 0 = (0_g‘𝐹))
50	45, 49	eqeq12d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (𝑦 , 𝑦) = (0_g‘𝐹)))
51	44, 50	bitr3d 284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦)) = 0 ↔ (𝑦 , 𝑦) = (0_g‘𝐹)))
52		eqid 2758	. . . . . . . . 9 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
53	14, 6, 5, 52, 9	ipeq0 20403	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0_g‘𝐹) ↔ 𝑦 = (0_g‘𝑊)))
54	1, 53	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0_g‘𝐹) ↔ 𝑦 = (0_g‘𝑊)))
55	27, 51, 54	3bitrd 308	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ 𝑦 = (0_g‘𝑊)))
56	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
57	33	simp1d 1139	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
58	57	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
59		3anass 1092	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
60		tcphcph.3	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)
61		simpr2 1192	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℝ)
62	61	recnd 10707	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℂ)
63	62	sqrtcld 14845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ ℂ)
64	60, 63	jca 515	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))
65	64	ex 416	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)))
66	34	eleq2d 2837	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ (𝐾 ∩ ℂ)))
67		recn 10665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
68		elin 3874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐾 ∩ ℂ) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℂ))
69	68	rbaib 542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾))
70	67, 69	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾))
71	66, 70	sylan9bb 513	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
72	71	adantrr 716	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
73	72	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾)))
74	73	pm5.32rd 581	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))))
75		3anass 1092	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
76	74, 75	bitr4di 292	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
77	34	eleq2d 2837	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ (√‘𝑥) ∈ (𝐾 ∩ ℂ)))
78		elin 3874	. . . . . . . . . . . . 13 ⊢ ((√‘𝑥) ∈ (𝐾 ∩ ℂ) ↔ ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))
79	77, 78	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)))
80	65, 76, 79	3imtr4d 297	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹)))
81	59, 80	syl5bi 245	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘𝑥) ∈ (Base‘𝐹)))
82	81	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
83	82	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
84	17	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
85		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
86		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
87	2, 5, 14, 56, 58, 6, 83, 84, 28, 8, 85, 86	tcphcphlem1 23935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))) ≤ ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
88	5, 8	grpsubcl 18246	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
89	88	3expb 1117	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
90	13, 89	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
91		oveq12 7159	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦(-_g‘𝑊)𝑧) ∧ 𝑥 = (𝑦(-_g‘𝑊)𝑧)) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
92	91	anidms 570	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
93	92	fveq2d 6662	. . . . . . . . 9 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (√‘(𝑥 , 𝑥)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
94	93, 23, 24	fvmpt3i 6764	. . . . . . . 8 ⊢ ((𝑦(-_g‘𝑊)𝑧) ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-_g‘𝑊)𝑧)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
95	90, 94	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-_g‘𝑊)𝑧)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
96		oveq12 7159	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑥 , 𝑥) = (𝑧 , 𝑧))
97	96	anidms 570	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 , 𝑥) = (𝑧 , 𝑧))
98	97	fveq2d 6662	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (√‘(𝑥 , 𝑥)) = (√‘(𝑧 , 𝑧)))
99	98, 23, 24	fvmpt3i 6764	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧) = (√‘(𝑧 , 𝑧)))
100	25, 99	oveqan12d 7169	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
101	100	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
102	87, 95, 101	3brtr4d 5064	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-_g‘𝑊)𝑧)) ≤ (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)))
103	7, 5, 8, 9, 13, 19, 55, 102	tngngpd 23355	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ NrmGrp)
104		phllmod 20395	. . . . . 6 ⊢ (𝐺 ∈ PreHil → 𝐺 ∈ LMod)
105	4, 104	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ LMod)
106		cnnrg 23482	. . . . . . 7 ⊢ ℂ_fld ∈ NrmRing
107	33	simp3d 1141	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubRing‘ℂ_fld))
108		eqid 2758	. . . . . . . 8 ⊢ (ℂ_fld ↾_s (Base‘𝐹)) = (ℂ_fld ↾_s (Base‘𝐹))
109	108	subrgnrg 23375	. . . . . . 7 ⊢ ((ℂ_fld ∈ NrmRing ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)) → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
110	106, 107, 109	sylancr 590	. . . . . 6 ⊢ (𝜑 → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
111	57, 110	eqeltrd 2852	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ NrmRing)
112	103, 105, 111	3jca 1125	. . . 4 ⊢ (𝜑 → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing))
113	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
114	57	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
115	82	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
116	17	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
117		eqid 2758	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
118		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝐹))
119		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
120	2, 5, 14, 113, 114, 6, 115, 116, 28, 117, 118, 119	tcphcphlem2 23936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦( ·_𝑠 ‘𝑊)𝑧) , (𝑦( ·_𝑠 ‘𝑊)𝑧))) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧))))
121	5, 14, 117, 28	lmodvscl 19719	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
122	121	3expb 1117	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
123	11, 122	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
124		eqid 2758	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
125	2, 124, 5, 6	tcphnmval 23929	. . . . . . 7 ⊢ ((𝑊 ∈ Grp ∧ (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (√‘((𝑦( ·_𝑠 ‘𝑊)𝑧) , (𝑦( ·_𝑠 ‘𝑊)𝑧))))
126	13, 123, 125	syl2an2r 684	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (√‘((𝑦( ·_𝑠 ‘𝑊)𝑧) , (𝑦( ·_𝑠 ‘𝑊)𝑧))))
127	114	fveq2d 6662	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (norm‘𝐹) = (norm‘(ℂ_fld ↾_s (Base‘𝐹))))
128	127	fveq1d 6660	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦))
129		subrgsubg 19609	. . . . . . . . . 10 ⊢ ((Base‘𝐹) ∈ (SubRing‘ℂ_fld) → (Base‘𝐹) ∈ (SubGrp‘ℂ_fld))
130	107, 129	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubGrp‘ℂ_fld))
131		cnfldnm 23480	. . . . . . . . . 10 ⊢ abs = (norm‘ℂ_fld)
132		eqid 2758	. . . . . . . . . 10 ⊢ (norm‘(ℂ_fld ↾_s (Base‘𝐹))) = (norm‘(ℂ_fld ↾_s (Base‘𝐹)))
133	108, 131, 132	subgnm2 23336	. . . . . . . . 9 ⊢ (((Base‘𝐹) ∈ (SubGrp‘ℂ_fld) ∧ 𝑦 ∈ (Base‘𝐹)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
134	130, 118, 133	syl2an2r 684	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
135	128, 134	eqtrd 2793	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = (abs‘𝑦))
136	2, 124, 5, 6	tcphnmval 23929	. . . . . . . 8 ⊢ ((𝑊 ∈ Grp ∧ 𝑧 ∈ 𝑉) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
137	13, 119, 136	syl2an2r 684	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
138	135, 137	oveq12d 7168	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧))))
139	120, 126, 138	3eqtr4d 2803	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
140	139	ralrimivva 3120	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
141	2, 5	tcphbas 23919	. . . . 5 ⊢ 𝑉 = (Base‘𝐺)
142	2, 117	tcphvsca 23924	. . . . 5 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝐺)
143	2, 14	tcphsca 23923	. . . . 5 ⊢ 𝐹 = (Scalar‘𝐺)
144		eqid 2758	. . . . 5 ⊢ (norm‘𝐹) = (norm‘𝐹)
145	141, 124, 142, 143, 28, 144	isnlm 23377	. . . 4 ⊢ (𝐺 ∈ NrmMod ↔ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧))))
146	112, 140, 145	sylanbrc 586	. . 3 ⊢ (𝜑 → 𝐺 ∈ NrmMod)
147	4, 146, 57	3jca 1125	. 2 ⊢ (𝜑 → (𝐺 ∈ PreHil ∧ 𝐺 ∈ NrmMod ∧ 𝐹 = (ℂ_fld ↾_s (Base‘𝐹))))
148		elin 3874	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)))
149		elrege0 12886	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
150	149	anbi2i 625	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
151	148, 150	bitri 278	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
152	151, 80	syl5bi 245	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) → (√‘𝑥) ∈ (Base‘𝐹)))
153	152	ralrimiv 3112	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹))
154		sqrtf 14771	. . . . 5 ⊢ √:ℂ⟶ℂ
155		ffun 6501	. . . . 5 ⊢ (√:ℂ⟶ℂ → Fun √)
156	154, 155	ax-mp 5	. . . 4 ⊢ Fun √
157		inss1 4133	. . . . . 6 ⊢ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ (Base‘𝐹)
158	157, 36	sstrid 3903	. . . . 5 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ ℂ)
159	154	fdmi 6509	. . . . 5 ⊢ dom √ = ℂ
160	158, 159	sseqtrrdi 3943	. . . 4 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √)
161		funimass4 6718	. . . 4 ⊢ ((Fun √ ∧ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √) → ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)))
162	156, 160, 161	sylancr 590	. . 3 ⊢ (𝜑 → ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)))
163	153, 162	mpbird 260	. 2 ⊢ (𝜑 → (√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹))
164	42	fmpttd 6870	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ)
165	2, 5, 6	tcphval 23918	. . . . 5 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
166		cnex 10656	. . . . 5 ⊢ ℂ ∈ V
167	165, 5, 166	tngnm 23353	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ) → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
168	13, 164, 167	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
169	168	eqcomd 2764	. 2 ⊢ (𝜑 → (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
170	2, 6	tcphip 23925	. . 3 ⊢ , = (·_𝑖‘𝐺)
171	141, 170, 124, 143, 28	iscph 23871	. 2 ⊢ (𝐺 ∈ ℂPreHil ↔ ((𝐺 ∈ PreHil ∧ 𝐺 ∈ NrmMod ∧ 𝐹 = (ℂ_fld ↾_s (Base‘𝐹))) ∧ (√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ∧ (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦)))))
172	147, 163, 169, 171	syl3anbrc 1340	1 ⊢ (𝜑 → 𝐺 ∈ ℂPreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∩ cin 3857 ⊆ wss 3858 class class class wbr 5032 ↦ cmpt 5112 dom cdm 5524 “ cima 5527 Fun wfun 6329 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 0cc0 10575 + caddc 10578 · cmul 10580 +∞cpnf 10710 ≤ cle 10714 2c2 11729 [,)cico 12781 ↑cexp 13479 √csqrt 14640 abscabs 14641 Basecbs 16541 ↾_s cress 16542 Scalarcsca 16626 ·_𝑠 cvsca 16627 ·_𝑖cip 16628 0_gc0g 16771 Grpcgrp 18169 -_gcsg 18171 SubGrpcsubg 18340 DivRingcdr 19570 SubRingcsubrg 19599 LModclmod 19702 LVecclvec 19942 ℂ_fldccnfld 20166 PreHilcphl 20389 normcnm 23278 NrmGrpcngp 23279 NrmRingcnrg 23281 NrmModcnlm 23282 ℂModcclm 23763 ℂPreHilccph 23867 toℂPreHilctcph 23868

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-tpos 7902 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-er 8299 df-map 8418 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-sup 8939 df-inf 8940 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ico 12785 df-fz 12940 df-seq 13419 df-exp 13480 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-rest 16754 df-topn 16755 df-0g 16773 df-topgen 16775 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-mhm 18022 df-grp 18172 df-minusg 18173 df-sbg 18174 df-subg 18343 df-ghm 18423 df-cmn 18975 df-abl 18976 df-mgp 19308 df-ur 19320 df-ring 19367 df-cring 19368 df-oppr 19444 df-dvdsr 19462 df-unit 19463 df-invr 19493 df-dvr 19504 df-rnghom 19538 df-drng 19572 df-subrg 19601 df-abv 19656 df-staf 19684 df-srng 19685 df-lmod 19704 df-lmhm 19862 df-lvec 19943 df-sra 20012 df-rgmod 20013 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-cnfld 20167 df-phl 20391 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-xms 23022 df-ms 23023 df-nm 23284 df-ngp 23285 df-tng 23286 df-nrg 23287 df-nlm 23288 df-clm 23764 df-cph 23869 df-tcph 23870

This theorem is referenced by: rrxcph 24092

W3C validator