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Theorem tcphcph 25144

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 25134	. . . 4 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)
4	1, 3	sylib 218	. . 3 ⊢ (𝜑 → 𝐺 ∈ PreHil)
5		tcphcph.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
6		tcphcph.h	. . . . . . 7 ⊢ , = (·_𝑖‘𝑊)
7	2, 5, 6	tcphval 25125	. . . . . 6 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
8		eqid 2730	. . . . . 6 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
9		eqid 2730	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
10		phllmod 21546	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
11	1, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
12		lmodgrp 20780	. . . . . . 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 25140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ)
17		tcphcph.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
18	16, 17	resqrtcld 15391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ ℝ)
19	18	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶ℝ)
20		oveq12 7399	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 , 𝑥) = (𝑦 , 𝑦))
21	20	anidms 566	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 , 𝑥) = (𝑦 , 𝑦))
22	21	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (√‘(𝑥 , 𝑥)) = (√‘(𝑦 , 𝑦)))
23		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
24		fvex 6874	. . . . . . . . . 10 ⊢ (√‘(𝑥 , 𝑥)) ∈ V
25	22, 23, 24	fvmpt3i 6976	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
26	25	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
27	26	eqeq1d 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
28		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐹) = (Base‘𝐹)
29		phllvec 21545	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
30	1, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ LVec)
31	14	lvecdrng 21019	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
32	30, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ DivRing)
33	28, 15, 32	cphsubrglem 25084	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 = (ℂ_fld ↾_s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)))
34	33	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (𝐾 ∩ ℂ))
35		inss2 4204	. . . . . . . . . . . . 13 ⊢ (𝐾 ∩ ℂ) ⊆ ℂ
36	34, 35	eqsstrdi 3994	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐹) ⊆ ℂ)
37	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (Base‘𝐹) ⊆ ℂ)
38	14, 6, 5, 28	ipcl 21549	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
39	38	3anidm23 1423	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
40	1, 39	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
41	37, 40	sseldd 3950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ ℂ)
42	41	sqrtcld 15413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (√‘(𝑦 , 𝑦)) ∈ ℂ)
43		sqeq0 14092	. . . . . . . . 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 15415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦))↑2) = (𝑦 , 𝑦))
46	2, 5, 14, 1, 15	phclm 25139	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
47	14	clm0 24979	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂMod → 0 = (0_g‘𝐹))
48	46, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (0_g‘𝐹))
49	48	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 0 = (0_g‘𝐹))
50	45, 49	eqeq12d 2746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (𝑦 , 𝑦) = (0_g‘𝐹)))
51	44, 50	bitr3d 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦)) = 0 ↔ (𝑦 , 𝑦) = (0_g‘𝐹)))
52		eqid 2730	. . . . . . . . 9 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
53	14, 6, 5, 52, 9	ipeq0 21554	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0_g‘𝐹) ↔ 𝑦 = (0_g‘𝑊)))
54	1, 53	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0_g‘𝐹) ↔ 𝑦 = (0_g‘𝑊)))
55	27, 51, 54	3bitrd 305	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ 𝑦 = (0_g‘𝑊)))
56	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
57	33	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
58	57	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
59		3anass 1094	. . . . . . . . . . 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 1196	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℝ)
62	61	recnd 11209	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℂ)
63	62	sqrtcld 15413	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ ℂ)
64	60, 63	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))
65	64	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)))
66	34	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ (𝐾 ∩ ℂ)))
67		recn 11165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
68		elin 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐾 ∩ ℂ) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℂ))
69	68	rbaib 538	. . . . . . . . . . . . . . . . . 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 509	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
72	71	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
73	72	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾)))
74	73	pm5.32rd 578	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))))
75		3anass 1094	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
76	74, 75	bitr4di 289	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
77	34	eleq2d 2815	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ (√‘𝑥) ∈ (𝐾 ∩ ℂ)))
78		elin 3933	. . . . . . . . . . . . 13 ⊢ ((√‘𝑥) ∈ (𝐾 ∩ ℂ) ↔ ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))
79	77, 78	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)))
80	65, 76, 79	3imtr4d 294	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹)))
81	59, 80	biimtrid 242	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘𝑥) ∈ (Base‘𝐹)))
82	81	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
83	82	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
84	17	adantlr 715	. . . . . . . 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 25142	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))) ≤ ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
88	5, 8	grpsubcl 18959	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
89	88	3expb 1120	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
90	13, 89	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
91		oveq12 7399	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦(-_g‘𝑊)𝑧) ∧ 𝑥 = (𝑦(-_g‘𝑊)𝑧)) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
92	91	anidms 566	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
93	92	fveq2d 6865	. . . . . . . . 9 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (√‘(𝑥 , 𝑥)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
94	93, 23, 24	fvmpt3i 6976	. . . . . . . 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 7399	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑥 , 𝑥) = (𝑧 , 𝑧))
97	96	anidms 566	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 , 𝑥) = (𝑧 , 𝑧))
98	97	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (√‘(𝑥 , 𝑥)) = (√‘(𝑧 , 𝑧)))
99	98, 23, 24	fvmpt3i 6976	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧) = (√‘(𝑧 , 𝑧)))
100	25, 99	oveqan12d 7409	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
101	100	adantl 481	. . . . . . 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 5142	. . . . . 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 24548	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ NrmGrp)
104		phllmod 21546	. . . . . 6 ⊢ (𝐺 ∈ PreHil → 𝐺 ∈ LMod)
105	4, 104	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ LMod)
106		cnnrg 24675	. . . . . . 7 ⊢ ℂ_fld ∈ NrmRing
107	33	simp3d 1144	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubRing‘ℂ_fld))
108		eqid 2730	. . . . . . . 8 ⊢ (ℂ_fld ↾_s (Base‘𝐹)) = (ℂ_fld ↾_s (Base‘𝐹))
109	108	subrgnrg 24568	. . . . . . 7 ⊢ ((ℂ_fld ∈ NrmRing ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)) → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
110	106, 107, 109	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
111	57, 110	eqeltrd 2829	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ NrmRing)
112	103, 105, 111	3jca 1128	. . . 4 ⊢ (𝜑 → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing))
113	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
114	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
115	82	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
116	17	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
117		eqid 2730	. . . . . . 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 25143	. . . . . 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 20791	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
122	121	3expb 1120	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
123	11, 122	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
124		eqid 2730	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
125	2, 124, 5, 6	tcphnmval 25136	. . . . . . 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 685	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (√‘((𝑦( ·_𝑠 ‘𝑊)𝑧) , (𝑦( ·_𝑠 ‘𝑊)𝑧))))
127	114	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (norm‘𝐹) = (norm‘(ℂ_fld ↾_s (Base‘𝐹))))
128	127	fveq1d 6863	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦))
129		subrgsubg 20493	. . . . . . . . . 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 24673	. . . . . . . . . 10 ⊢ abs = (norm‘ℂ_fld)
132		eqid 2730	. . . . . . . . . 10 ⊢ (norm‘(ℂ_fld ↾_s (Base‘𝐹))) = (norm‘(ℂ_fld ↾_s (Base‘𝐹)))
133	108, 131, 132	subgnm2 24529	. . . . . . . . 9 ⊢ (((Base‘𝐹) ∈ (SubGrp‘ℂ_fld) ∧ 𝑦 ∈ (Base‘𝐹)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
134	130, 118, 133	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
135	128, 134	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = (abs‘𝑦))
136	2, 124, 5, 6	tcphnmval 25136	. . . . . . . 8 ⊢ ((𝑊 ∈ Grp ∧ 𝑧 ∈ 𝑉) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
137	13, 119, 136	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
138	135, 137	oveq12d 7408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧))))
139	120, 126, 138	3eqtr4d 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
140	139	ralrimivva 3181	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
141	2, 5	tcphbas 25126	. . . . 5 ⊢ 𝑉 = (Base‘𝐺)
142	2, 117	tcphvsca 25131	. . . . 5 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝐺)
143	2, 14	tcphsca 25130	. . . . 5 ⊢ 𝐹 = (Scalar‘𝐺)
144		eqid 2730	. . . . 5 ⊢ (norm‘𝐹) = (norm‘𝐹)
145	141, 124, 142, 143, 28, 144	isnlm 24570	. . . 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 583	. . 3 ⊢ (𝜑 → 𝐺 ∈ NrmMod)
147	4, 146, 57	3jca 1128	. 2 ⊢ (𝜑 → (𝐺 ∈ PreHil ∧ 𝐺 ∈ NrmMod ∧ 𝐹 = (ℂ_fld ↾_s (Base‘𝐹))))
148		elin 3933	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)))
149		elrege0 13422	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
150	149	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
151	148, 150	bitri 275	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))
152	151, 80	biimtrid 242	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) → (√‘𝑥) ∈ (Base‘𝐹)))
153	152	ralrimiv 3125	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹))
154		sqrtf 15337	. . . . 5 ⊢ √:ℂ⟶ℂ
155		ffun 6694	. . . . 5 ⊢ (√:ℂ⟶ℂ → Fun √)
156	154, 155	ax-mp 5	. . . 4 ⊢ Fun √
157		inss1 4203	. . . . . 6 ⊢ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ (Base‘𝐹)
158	157, 36	sstrid 3961	. . . . 5 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ ℂ)
159	154	fdmi 6702	. . . . 5 ⊢ dom √ = ℂ
160	158, 159	sseqtrrdi 3991	. . . 4 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √)
161		funimass4 6928	. . . 4 ⊢ ((Fun √ ∧ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √) → ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)))
162	156, 160, 161	sylancr 587	. . 3 ⊢ (𝜑 → ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)))
163	153, 162	mpbird 257	. 2 ⊢ (𝜑 → (√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹))
164	42	fmpttd 7090	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ)
165	2, 5, 6	tcphval 25125	. . . . 5 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
166		cnex 11156	. . . . 5 ⊢ ℂ ∈ V
167	165, 5, 166	tngnm 24546	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ) → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
168	13, 164, 167	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
169	168	eqcomd 2736	. 2 ⊢ (𝜑 → (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
170	2, 6	tcphip 25132	. . 3 ⊢ , = (·_𝑖‘𝐺)
171	141, 170, 124, 143, 28	iscph 25077	. 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 1344	1 ⊢ (𝜑 → 𝐺 ∈ ℂPreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∩ cin 3916 ⊆ wss 3917 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 “ cima 5644 Fun wfun 6508 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 · cmul 11080 +∞cpnf 11212 ≤ cle 11216 2c2 12248 [,)cico 13315 ↑cexp 14033 √csqrt 15206 abscabs 15207 Basecbs 17186 ↾_s cress 17207 Scalarcsca 17230 ·_𝑠 cvsca 17231 ·_𝑖cip 17232 0_gc0g 17409 Grpcgrp 18872 -_gcsg 18874 SubGrpcsubg 19059 SubRingcsubrg 20485 DivRingcdr 20645 LModclmod 20773 LVecclvec 21016 ℂ_fldccnfld 21271 PreHilcphl 21540 normcnm 24471 NrmGrpcngp 24472 NrmRingcnrg 24474 NrmModcnlm 24475 ℂModcclm 24969 ℂPreHilccph 25073 toℂPreHilctcph 25074

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-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

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-iun 4960 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-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-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 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-ico 13319 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 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-rest 17392 df-topn 17393 df-0g 17411 df-topgen 17413 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-ghm 19152 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-abv 20725 df-staf 20755 df-srng 20756 df-lmod 20775 df-lmhm 20936 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-phl 21542 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-xms 24215 df-ms 24216 df-nm 24477 df-ngp 24478 df-tng 24479 df-nrg 24480 df-nlm 24481 df-clm 24970 df-cph 25075 df-tcph 25076

This theorem is referenced by: rrxcph 25299

W3C validator