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Theorem tcphcph 25535

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 25525	. . . 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 25516	. . . . . 6 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
8		eqid 2761	. . . . . 6 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
9		eqid 2761	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
10		phllmod 21913	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
11	1, 10	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
12		lmodgrp 21119	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
13	11, 12	syl 18	. . . . . 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 25531	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ)
17		tcphcph.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
18	16, 17	resqrtcld 15562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ ℝ)
19	18	fmpttd 7107	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶ℝ)
20		oveq12 7421	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 , 𝑥) = (𝑦 , 𝑦))
21	20	anidms 577	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 , 𝑥) = (𝑦 , 𝑦))
22	21	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (√‘(𝑥 , 𝑥)) = (√‘(𝑦 , 𝑦)))
23		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))
24		fvex 6890	. . . . . . . . . 10 ⊢ (√‘(𝑥 , 𝑥)) ∈ V
25	22, 23, 24	fvmpt3i 6991	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
26	25	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦)))
27	26	eqeq1d 2763	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
28		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐹) = (Base‘𝐹)
29		phllvec 21912	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
30	1, 29	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ LVec)
31	14	lvecdrng 21357	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
32	30, 31	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ DivRing)
33	28, 15, 32	cphsubrglem 25475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 = (ℂ_fld ↾_s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)))
34	33	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐹) = (𝐾 ∩ ℂ))
35		inss2 4183	. . . . . . . . . . . . 13 ⊢ (𝐾 ∩ ℂ) ⊆ ℂ
36	34, 35	eqsstrdi 3975	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐹) ⊆ ℂ)
37	36	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (Base‘𝐹) ⊆ ℂ)
38	14, 6, 5, 28	ipcl 21916	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
39	38	3anidm23 1448	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
40	1, 39	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹))
41	37, 40	sseldd 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ ℂ)
42	41	sqrtcld 15584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (√‘(𝑦 , 𝑦)) ∈ ℂ)
43		sqeq0 14240	. . . . . . . . 9 ⊢ ((√‘(𝑦 , 𝑦)) ∈ ℂ → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
44	42, 43	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0))
45	41	sqsqrtd 15586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦))↑2) = (𝑦 , 𝑦))
46	2, 5, 14, 1, 15	phclm 25530	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
47	14	clm0 25370	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂMod → 0 = (0_g‘𝐹))
48	46, 47	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (0_g‘𝐹))
49	48	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 0 = (0_g‘𝐹))
50	45, 49	eqeq12d 2777	. . . . . . . 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 2761	. . . . . . . . 9 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
53	14, 6, 5, 52, 9	ipeq0 21921	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0_g‘𝐹) ↔ 𝑦 = (0_g‘𝑊)))
54	1, 53	sylan 592	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
57	33	simp1d 1160	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
58	57	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
59		3anass 1111	. . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℝ)
62	61	recnd 11315	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℂ)
63	62	sqrtcld 15584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ ℂ)
64	60, 63	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))
65	64	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)))
66	34	eleq2d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ (𝐾 ∩ ℂ)))
67		recn 11268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
68		elin 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐾 ∩ ℂ) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℂ))
69	68	rbaib 548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾))
70	67, 69	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾))
71	66, 70	sylan9bb 519	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
72	71	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))
73	72	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾)))
74	73	pm5.32rd 589	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))))
75		3anass 1111	. . . . . . . . . . . . 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 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ (√‘𝑥) ∈ (𝐾 ∩ ℂ)))
78		elin 3915	. . . . . . . . . . . . 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	biimtrid 245	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘𝑥) ∈ (Base‘𝐹)))
82	81	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
83	82	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
84	17	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
85		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
86		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
87	2, 5, 14, 56, 58, 6, 83, 84, 28, 8, 85, 86	tcphcphlem1 25533	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))) ≤ ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
88	5, 8	grpsubcl 19207	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
89	88	3expb 1138	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
90	13, 89	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-_g‘𝑊)𝑧) ∈ 𝑉)
91		oveq12 7421	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦(-_g‘𝑊)𝑧) ∧ 𝑥 = (𝑦(-_g‘𝑊)𝑧)) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
92	91	anidms 577	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (𝑥 , 𝑥) = ((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧)))
93	92	fveq2d 6881	. . . . . . . . 9 ⊢ (𝑥 = (𝑦(-_g‘𝑊)𝑧) → (√‘(𝑥 , 𝑥)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
94	93, 23, 24	fvmpt3i 6991	. . . . . . . 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 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-_g‘𝑊)𝑧)) = (√‘((𝑦(-_g‘𝑊)𝑧) , (𝑦(-_g‘𝑊)𝑧))))
96		oveq12 7421	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑥 , 𝑥) = (𝑧 , 𝑧))
97	96	anidms 577	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 , 𝑥) = (𝑧 , 𝑧))
98	97	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (√‘(𝑥 , 𝑥)) = (√‘(𝑧 , 𝑧)))
99	98, 23, 24	fvmpt3i 6991	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧) = (√‘(𝑧 , 𝑧)))
100	25, 99	oveqan12d 7431	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧))))
101	100	adantl 487	. . . . . . 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 5137	. . . . . 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 24949	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ NrmGrp)
104		phllmod 21913	. . . . . 6 ⊢ (𝐺 ∈ PreHil → 𝐺 ∈ LMod)
105	4, 104	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ LMod)
106		cnnrg 25076	. . . . . . 7 ⊢ ℂ_fld ∈ NrmRing
107	33	simp3d 1162	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubRing‘ℂ_fld))
108		eqid 2761	. . . . . . . 8 ⊢ (ℂ_fld ↾_s (Base‘𝐹)) = (ℂ_fld ↾_s (Base‘𝐹))
109	108	subrgnrg 24969	. . . . . . 7 ⊢ ((ℂ_fld ∈ NrmRing ∧ (Base‘𝐹) ∈ (SubRing‘ℂ_fld)) → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
110	106, 107, 109	sylancr 599	. . . . . 6 ⊢ (𝜑 → (ℂ_fld ↾_s (Base‘𝐹)) ∈ NrmRing)
111	57, 110	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ NrmRing)
112	103, 105, 111	3jca 1146	. . . 4 ⊢ (𝜑 → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing))
113	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil)
114	57	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂ_fld ↾_s (Base‘𝐹)))
115	82	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))
116	17	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
117		eqid 2761	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
118		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝐹))
119		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
120	2, 5, 14, 113, 114, 6, 115, 116, 28, 117, 118, 119	tcphcphlem2 25534	. . . . . 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 21130	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
122	121	3expb 1138	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
123	11, 122	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
124		eqid 2761	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
125	2, 124, 5, 6	tcphnmval 25527	. . . . . . 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 698	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (√‘((𝑦( ·_𝑠 ‘𝑊)𝑧) , (𝑦( ·_𝑠 ‘𝑊)𝑧))))
127	114	fveq2d 6881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (norm‘𝐹) = (norm‘(ℂ_fld ↾_s (Base‘𝐹))))
128	127	fveq1d 6879	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦))
129		subrgsubg 20806	. . . . . . . . . 10 ⊢ ((Base‘𝐹) ∈ (SubRing‘ℂ_fld) → (Base‘𝐹) ∈ (SubGrp‘ℂ_fld))
130	107, 129	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubGrp‘ℂ_fld))
131		cnfldnm 25074	. . . . . . . . . 10 ⊢ abs = (norm‘ℂ_fld)
132		eqid 2761	. . . . . . . . . 10 ⊢ (norm‘(ℂ_fld ↾_s (Base‘𝐹))) = (norm‘(ℂ_fld ↾_s (Base‘𝐹)))
133	108, 131, 132	subgnm2 24930	. . . . . . . . 9 ⊢ (((Base‘𝐹) ∈ (SubGrp‘ℂ_fld) ∧ 𝑦 ∈ (Base‘𝐹)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
134	130, 118, 133	syl2an2r 698	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘(ℂ_fld ↾_s (Base‘𝐹)))‘𝑦) = (abs‘𝑦))
135	128, 134	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = (abs‘𝑦))
136	2, 124, 5, 6	tcphnmval 25527	. . . . . . . 8 ⊢ ((𝑊 ∈ Grp ∧ 𝑧 ∈ 𝑉) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
137	13, 119, 136	syl2an2r 698	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧)))
138	135, 137	oveq12d 7430	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧))))
139	120, 126, 138	3eqtr4d 2806	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
140	139	ralrimivva 3206	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·_𝑠 ‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))
141	2, 5	tcphbas 25517	. . . . 5 ⊢ 𝑉 = (Base‘𝐺)
142	2, 117	tcphvsca 25522	. . . . 5 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝐺)
143	2, 14	tcphsca 25521	. . . . 5 ⊢ 𝐹 = (Scalar‘𝐺)
144		eqid 2761	. . . . 5 ⊢ (norm‘𝐹) = (norm‘𝐹)
145	141, 124, 142, 143, 28, 144	isnlm 24971	. . . 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 595	. . 3 ⊢ (𝜑 → 𝐺 ∈ NrmMod)
147	4, 146, 57	3jca 1146	. 2 ⊢ (𝜑 → (𝐺 ∈ PreHil ∧ 𝐺 ∈ NrmMod ∧ 𝐹 = (ℂ_fld ↾_s (Base‘𝐹))))
148		elin 3915	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)))
149		elrege0 13563	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
150	149	anbi2i 635	. . . . . 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	biimtrid 245	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) → (√‘𝑥) ∈ (Base‘𝐹)))
153	152	ralrimiv 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹))
154		sqrtf 15508	. . . . 5 ⊢ √:ℂ⟶ℂ
155		ffun 6704	. . . . 5 ⊢ (√:ℂ⟶ℂ → Fun √)
156	154, 155	ax-mp 5	. . . 4 ⊢ Fun √
157		inss1 4182	. . . . . 6 ⊢ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ (Base‘𝐹)
158	157, 36	sstrid 3942	. . . . 5 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ ℂ)
159	154	fdmi 6713	. . . . 5 ⊢ dom √ = ℂ
160	158, 159	sseqtrrdi 3972	. . . 4 ⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √)
161		funimass4 6941	. . . 4 ⊢ ((Fun √ ∧ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √) → ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)))
162	156, 160, 161	sylancr 599	. . 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 7107	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ)
165	2, 5, 6	tcphval 25516	. . . . 5 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
166		cnex 11259	. . . . 5 ⊢ ℂ ∈ V
167	165, 5, 166	tngnm 24947	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ) → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
168	13, 164, 167	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺))
169	168	eqcomd 2767	. 2 ⊢ (𝜑 → (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))
170	2, 6	tcphip 25523	. . 3 ⊢ , = (·_𝑖‘𝐺)
171	141, 170, 124, 143, 28	iscph 25468	. 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 1362	1 ⊢ (𝜑 → 𝐺 ∈ ℂPreHil)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 “ cima 5654 Fun wfun 6525 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ℂcc 11176 ℝcr 11177 0cc0 11178 + caddc 11181 · cmul 11183 +∞cpnf 11318 ≤ cle 11322 2c2 12375 [,)cico 13456 ↑cexp 14181 √csqrt 15377 abscabs 15378 Basecbs 17364 ↾_s cress 17385 Scalarcsca 17408 ·_𝑠 cvsca 17409 ·_𝑖cip 17410 0_gc0g 17587 Grpcgrp 19121 -_gcsg 19123 SubGrpcsubg 19307 SubRingcsubrg 20798 DivRingcdr 20957 LModclmod 21112 LVecclvec 21354 ℂ_fldccnfld 21655 PreHilcphl 21907 normcnm 24872 NrmGrpcngp 24873 NrmRingcnrg 24875 NrmModcnlm 24876 ℂModcclm 25360 ℂPreHilccph 25464 toℂPreHilctcph 25465

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11234 ax-resscn 11235 ax-1cn 11236 ax-icn 11237 ax-addcl 11238 ax-addrcl 11239 ax-mulcl 11240 ax-mulrcl 11241 ax-mulcom 11242 ax-addass 11243 ax-mulass 11244 ax-distr 11245 ax-i2m1 11246 ax-1ne0 11247 ax-1rid 11248 ax-rnegex 11249 ax-rrecex 11250 ax-cnre 11251 ax-pre-lttri 11252 ax-pre-lttrn 11253 ax-pre-ltadd 11254 ax-pre-mulgt0 11255 ax-pre-sup 11256 ax-addf 11257 ax-mulf 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9418 df-inf 9419 df-pnf 11323 df-mnf 11324 df-xr 11325 df-ltxr 11326 df-le 11327 df-sub 11521 df-neg 11522 df-div 11952 df-nn 12314 df-2 12383 df-3 12384 df-4 12385 df-5 12386 df-6 12387 df-7 12388 df-8 12389 df-9 12390 df-n0 12585 df-z 12672 df-dec 12793 df-uz 12944 df-q 13054 df-rp 13099 df-xneg 13219 df-xadd 13220 df-xmul 13221 df-ico 13460 df-fz 13618 df-seq 14122 df-exp 14182 df-cj 15243 df-re 15244 df-im 15245 df-sqrt 15379 df-abs 15380 df-struct 17302 df-sets 17319 df-slot 17337 df-ndx 17349 df-base 17365 df-ress 17386 df-plusg 17418 df-mulr 17419 df-starv 17420 df-sca 17421 df-vsca 17422 df-ip 17423 df-tset 17424 df-ple 17425 df-ds 17427 df-unif 17428 df-rest 17570 df-topn 17571 df-0g 17589 df-topgen 17591 df-mgm 18793 df-sgrp 18885 df-mnd 18901 df-mhm 18955 df-grp 19124 df-minusg 19125 df-sbg 19126 df-subg 19310 df-ghm 19405 df-cmn 19973 df-abl 19974 df-mgp 20338 df-rng 20352 df-ur 20385 df-ring 20438 df-cring 20439 df-oppr 20544 df-dvdsr 20564 df-unit 20565 df-invr 20595 df-dvr 20608 df-rhm 20679 df-subrng 20775 df-subrg 20799 df-drng 20959 df-abv 21043 df-staf 21073 df-srng 21074 df-lmod 21114 df-lmhm 21274 df-lvec 21355 df-sra 21425 df-rgmod 21426 df-psmet 21647 df-xmet 21648 df-met 21649 df-bl 21650 df-mopn 21651 df-cnfld 21656 df-phl 21909 df-top 23189 df-topon 23206 df-topsp 23228 df-bases 23241 df-xms 24616 df-ms 24617 df-nm 24878 df-ngp 24879 df-tng 24880 df-nrg 24881 df-nlm 24882 df-clm 25361 df-cph 25466 df-tcph 25467

This theorem is used by: rrxcph 25690

W3C validator