ipcau2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ipcau2		Structured version Visualization version GIF version

Theorem ipcau2 24751

Description: The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space built from a pre-Hilbert space with certain properties. The main theorem is ipcau 24755. (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))
tcphcph.k	⊢ 𝐾 = (Base‘𝐹)
ipcau2.n	⊢ 𝑁 = (norm‘𝐺)
ipcau2.c	⊢ 𝐶 = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
ipcau2.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
ipcau2.4	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
ipcau2	⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌)))

Distinct variable groups: 𝑥, , 𝑥,𝐹 𝑥,𝐺 𝑥,𝑉 𝑥,𝐶 𝜑,𝑥 𝑥,𝑊 𝑥,𝑋 𝑥,𝑌 Allowed substitution hints: 𝐾(𝑥) 𝑁(𝑥)

**Proof of Theorem ipcau2**
Step	Hyp	Ref	Expression
1		oveq2 7417	. . . . . . 7 ⊢ (𝑌 = (0_g‘𝑊) → (𝑋 , 𝑌) = (𝑋 , (0_g‘𝑊)))
2	1	oveq1d 7424	. . . . . 6 ⊢ (𝑌 = (0_g‘𝑊) → ((𝑋 , 𝑌) · (𝑌 , 𝑋)) = ((𝑋 , (0_g‘𝑊)) · (𝑌 , 𝑋)))
3	2	breq1d 5159	. . . . 5 ⊢ (𝑌 = (0_g‘𝑊) → (((𝑋 , 𝑌) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌)) ↔ ((𝑋 , (0_g‘𝑊)) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌))))
4		tcphval.n	. . . . . . . . . . . . 13 ⊢ 𝐺 = (toℂPreHil‘𝑊)
5		tcphcph.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (Base‘𝑊)
6		tcphcph.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (Scalar‘𝑊)
7		tcphcph.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ PreHil)
8		tcphcph.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
9	4, 5, 6, 7, 8	phclm 24749	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
10		tcphcph.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘𝐹)
11	6, 10	clmsscn 24595	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
12	9, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
13		ipcau2.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14		ipcau2.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		tcphcph.h	. . . . . . . . . . . . 13 ⊢ , = (·_𝑖‘𝑊)
16	6, 15, 5, 10	ipcl 21186	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾)
17	7, 13, 14, 16	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾)
18	12, 17	sseldd 3984	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
19	18	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑌) ∈ ℂ)
20	6, 15, 5, 10	ipcl 21186	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾)
21	7, 14, 13, 20	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾)
22	12, 21	sseldd 3984	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
23	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑋) ∈ ℂ)
24	4, 5, 6, 7, 8, 15	tcphcphlem3 24750	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
25	14, 24	mpdan 686	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
26	25	recnd 11242	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
27	26	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑌) ∈ ℂ)
28	6	clm0 24588	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂMod → 0 = (0_g‘𝐹))
29	9, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 = (0_g‘𝐹))
30	29	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌 , 𝑌) = 0 ↔ (𝑌 , 𝑌) = (0_g‘𝐹)))
31		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
32		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
33	6, 15, 5, 31, 32	ipeq0 21191	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉) → ((𝑌 , 𝑌) = (0_g‘𝐹) ↔ 𝑌 = (0_g‘𝑊)))
34	7, 14, 33	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌 , 𝑌) = (0_g‘𝐹) ↔ 𝑌 = (0_g‘𝑊)))
35	30, 34	bitrd 279	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌 , 𝑌) = 0 ↔ 𝑌 = (0_g‘𝑊)))
36	35	necon3bid 2986	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑌 , 𝑌) ≠ 0 ↔ 𝑌 ≠ (0_g‘𝑊)))
37	36	biimpar 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑌) ≠ 0)
38	19, 23, 27, 37	divassd 12025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) = ((𝑋 , 𝑌) · ((𝑌 , 𝑋) / (𝑌 , 𝑌))))
39		ipcau2.c	. . . . . . . . 9 ⊢ 𝐶 = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
40	39	oveq2i 7420	. . . . . . . 8 ⊢ ((𝑋 , 𝑌) · 𝐶) = ((𝑋 , 𝑌) · ((𝑌 , 𝑋) / (𝑌 , 𝑌)))
41	38, 40	eqtr4di 2791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) = ((𝑋 , 𝑌) · 𝐶))
42		oveq12 7418	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) ∧ 𝑥 = (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))) → (𝑥 , 𝑥) = ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))))
43	42	anidms 568	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) → (𝑥 , 𝑥) = ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))))
44	43	breq2d 5161	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))))
45		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
46	45	ralrimiva 3147	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
47	46	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
48		phllmod 21183	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
49	7, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ LMod)
50	49	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝑊 ∈ LMod)
51	13	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝑋 ∈ 𝑉)
52	39	fveq2i 6895	. . . . . . . . . . . . . . 15 ⊢ (∗‘𝐶) = (∗‘((𝑌 , 𝑋) / (𝑌 , 𝑌)))
53	23, 27, 37	cjdivd 15170	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘((𝑌 , 𝑋) / (𝑌 , 𝑌))) = ((∗‘(𝑌 , 𝑋)) / (∗‘(𝑌 , 𝑌))))
54	52, 53	eqtrid 2785	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘𝐶) = ((∗‘(𝑌 , 𝑋)) / (∗‘(𝑌 , 𝑌))))
55	8	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (_𝑟‘𝐹) = (_𝑟‘(ℂ_fld ↾_s 𝐾)))
56	10	fvexi 6906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
57		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
58		cnfldcj 20951	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∗ = (*_𝑟‘ℂ_fld)
59	57, 58	ressstarv 17253	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ V → ∗ = (*_𝑟‘(ℂ_fld ↾_s 𝐾)))
60	56, 59	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∗ = (*_𝑟‘(ℂ_fld ↾_s 𝐾))
61	55, 60	eqtr4di 2791	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (*_𝑟‘𝐹) = ∗)
62	61	fveq1d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (∗‘(𝑋 , 𝑌)))
63		eqid 2733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
64	6, 15, 5, 63	ipcj 21187	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
65	7, 13, 14, 64	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
66	62, 65	eqtr3d 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
67	66	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
68	67	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘(∗‘(𝑋 , 𝑌))) = (∗‘(𝑌 , 𝑋)))
69	19	cjcjd 15146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘(∗‘(𝑋 , 𝑌))) = (𝑋 , 𝑌))
70	68, 69	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘(𝑌 , 𝑋)) = (𝑋 , 𝑌))
71	25	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑌) ∈ ℝ)
72	71	cjred 15173	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘(𝑌 , 𝑌)) = (𝑌 , 𝑌))
73	70, 72	oveq12d 7427	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘(𝑌 , 𝑋)) / (∗‘(𝑌 , 𝑌))) = ((𝑋 , 𝑌) / (𝑌 , 𝑌)))
74	19, 27, 37	divrecd 11993	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) / (𝑌 , 𝑌)) = ((𝑋 , 𝑌) · (1 / (𝑌 , 𝑌))))
75	54, 73, 74	3eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘𝐶) = ((𝑋 , 𝑌) · (1 / (𝑌 , 𝑌))))
76	9	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝑊 ∈ ℂMod)
77	17	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑌) ∈ 𝐾)
78	6, 15, 5, 10	ipcl 21186	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾)
79	7, 14, 14, 78	syl3anc 1372	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾)
80	79	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑌) ∈ 𝐾)
81	8	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝐹 = (ℂ_fld ↾_s 𝐾))
82		phllvec 21182	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
83	7, 82	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ LVec)
84	6	lvecdrng 20716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
85	83, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ DivRing)
86	85	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝐹 ∈ DivRing)
87	10, 81, 86	cphreccllem 24695	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) ∧ (𝑌 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑌) ≠ 0) → (1 / (𝑌 , 𝑌)) ∈ 𝐾)
88	80, 37, 87	mpd3an23 1464	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (1 / (𝑌 , 𝑌)) ∈ 𝐾)
89	6, 10	clmmcl 24601	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (1 / (𝑌 , 𝑌)) ∈ 𝐾) → ((𝑋 , 𝑌) · (1 / (𝑌 , 𝑌))) ∈ 𝐾)
90	76, 77, 88, 89	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · (1 / (𝑌 , 𝑌))) ∈ 𝐾)
91	75, 90	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘𝐶) ∈ 𝐾)
92	14	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝑌 ∈ 𝑉)
93		eqid 2733	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
94	5, 6, 93, 10	lmodvscl 20489	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (∗‘𝐶) ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)
95	50, 91, 92, 94	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)
96		eqid 2733	. . . . . . . . . . . 12 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
97	5, 96	lmodvsubcl 20517	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉) → (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) ∈ 𝑉)
98	50, 51, 95, 97	syl3anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) ∈ 𝑉)
99	44, 47, 98	rspcdva 3614	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 0 ≤ ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))))
100		eqid 2733	. . . . . . . . . . 11 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
101		eqid 2733	. . . . . . . . . . 11 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
102	7	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝑊 ∈ PreHil)
103	6, 15, 5, 96, 100, 101, 102, 51, 95, 51, 95	ip2subdi 21197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))) = (((𝑋 , 𝑋)(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))(-_g‘𝐹)((𝑋 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋))))
104	81	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (+_g‘𝐹) = (+_g‘(ℂ_fld ↾_s 𝐾)))
105		cnfldadd 20949	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘ℂ_fld)
106	57, 105	ressplusg 17235	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ V → + = (+_g‘(ℂ_fld ↾_s 𝐾)))
107	56, 106	ax-mp 5	. . . . . . . . . . . . 13 ⊢ + = (+_g‘(ℂ_fld ↾_s 𝐾))
108	104, 107	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (+_g‘𝐹) = + )
109		eqidd 2734	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑋) = (𝑋 , 𝑋))
110		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
111	6, 15, 5, 10, 93, 110	ipass 21198	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ ((∗‘𝐶) ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = ((∗‘𝐶)(._r‘𝐹)(𝑌 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))))
112	102, 91, 92, 95, 111	syl13anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = ((∗‘𝐶)(._r‘𝐹)(𝑌 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))))
113	81	fveq2d 6896	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (._r‘𝐹) = (._r‘(ℂ_fld ↾_s 𝐾)))
114		cnfldmul 20950	. . . . . . . . . . . . . . . . 17 ⊢ · = (._r‘ℂ_fld)
115	57, 114	ressmulr 17252	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ V → · = (._r‘(ℂ_fld ↾_s 𝐾)))
116	56, 115	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ · = (._r‘(ℂ_fld ↾_s 𝐾))
117	113, 116	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (._r‘𝐹) = · )
118		eqidd 2734	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (∗‘𝐶) = (∗‘𝐶))
119	23, 27, 37	divrecd 11993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) · (1 / (𝑌 , 𝑌))))
120	39, 119	eqtrid 2785	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝐶 = ((𝑌 , 𝑋) · (1 / (𝑌 , 𝑌))))
121	21	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , 𝑋) ∈ 𝐾)
122	6, 10	clmmcl 24601	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑌 , 𝑋) ∈ 𝐾 ∧ (1 / (𝑌 , 𝑌)) ∈ 𝐾) → ((𝑌 , 𝑋) · (1 / (𝑌 , 𝑌))) ∈ 𝐾)
123	76, 121, 88, 122	syl3anc 1372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑋) · (1 / (𝑌 , 𝑌))) ∈ 𝐾)
124	120, 123	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝐶 ∈ 𝐾)
125	6, 15, 5, 10, 93, 110, 63	ipassr2 21200	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ (𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝑌 , 𝑌)(._r‘𝐹)𝐶) = (𝑌 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
126	102, 92, 92, 124, 125	syl13anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑌)(._r‘𝐹)𝐶) = (𝑌 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
127	117	oveqd 7426	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑌)(._r‘𝐹)𝐶) = ((𝑌 , 𝑌) · 𝐶))
128	39	oveq2i 7420	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 , 𝑌) · 𝐶) = ((𝑌 , 𝑌) · ((𝑌 , 𝑋) / (𝑌 , 𝑌)))
129	23, 27, 37	divcan2d 11992	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑌) · ((𝑌 , 𝑋) / (𝑌 , 𝑌))) = (𝑌 , 𝑋))
130	128, 129	eqtrid 2785	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑌) · 𝐶) = (𝑌 , 𝑋))
131	127, 130	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑌 , 𝑌)(._r‘𝐹)𝐶) = (𝑌 , 𝑋))
132	61	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (*_𝑟‘𝐹) = ∗)
133	132	fveq1d 6894	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((*_𝑟‘𝐹)‘𝐶) = (∗‘𝐶))
134	133	oveq1d 7424	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌) = ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))
135	134	oveq2d 7425	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = (𝑌 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
136	126, 131, 135	3eqtr3rd 2782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑌 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = (𝑌 , 𝑋))
137	117, 118, 136	oveq123d 7430	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘𝐶)(._r‘𝐹)(𝑌 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))) = ((∗‘𝐶) · (𝑌 , 𝑋)))
138	112, 137	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = ((∗‘𝐶) · (𝑌 , 𝑋)))
139	108, 109, 138	oveq123d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑋)(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))) = ((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))))
140	6, 15, 5, 10, 93, 110, 63	ipassr2 21200	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝑋 , 𝑌)(._r‘𝐹)𝐶) = (𝑋 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
141	102, 51, 92, 124, 140	syl13anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌)(._r‘𝐹)𝐶) = (𝑋 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
142	117	oveqd 7426	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌)(._r‘𝐹)𝐶) = ((𝑋 , 𝑌) · 𝐶))
143	134	oveq2d 7425	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , (((*_𝑟‘𝐹)‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = (𝑋 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))
144	141, 142, 143	3eqtr3rd 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) = ((𝑋 , 𝑌) · 𝐶))
145	6, 15, 5, 10, 93, 110	ipass 21198	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ ((∗‘𝐶) ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋) = ((∗‘𝐶)(._r‘𝐹)(𝑌 , 𝑋)))
146	102, 91, 92, 51, 145	syl13anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋) = ((∗‘𝐶)(._r‘𝐹)(𝑌 , 𝑋)))
147	117	oveqd 7426	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘𝐶)(._r‘𝐹)(𝑌 , 𝑋)) = ((∗‘𝐶) · (𝑌 , 𝑋)))
148	146, 147	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋) = ((∗‘𝐶) · (𝑌 , 𝑋)))
149	108, 144, 148	oveq123d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋)) = (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋))))
150	139, 149	oveq12d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑋)(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)))(-_g‘𝐹)((𝑋 , ((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))(+_g‘𝐹)(((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌) , 𝑋))) = (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋)))(-_g‘𝐹)(((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))))
151	6, 15, 5, 10	ipcl 21186	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾)
152	102, 51, 51, 151	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑋) ∈ 𝐾)
153	6, 10	clmmcl 24601	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ ℂMod ∧ (∗‘𝐶) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((∗‘𝐶) · (𝑌 , 𝑋)) ∈ 𝐾)
154	76, 91, 121, 153	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘𝐶) · (𝑌 , 𝑋)) ∈ 𝐾)
155	6, 10	clmacl 24600	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ ((∗‘𝐶) · (𝑌 , 𝑋)) ∈ 𝐾) → ((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾)
156	76, 152, 154, 155	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾)
157	6, 10	clmmcl 24601	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ 𝐶 ∈ 𝐾) → ((𝑋 , 𝑌) · 𝐶) ∈ 𝐾)
158	76, 77, 124, 157	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · 𝐶) ∈ 𝐾)
159	6, 10	clmacl 24600	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ ℂMod ∧ ((𝑋 , 𝑌) · 𝐶) ∈ 𝐾 ∧ ((∗‘𝐶) · (𝑌 , 𝑋)) ∈ 𝐾) → (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾)
160	76, 158, 154, 159	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾)
161	6, 10	clmsub 24596	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂMod ∧ ((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾 ∧ (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋))) ∈ 𝐾) → (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) − (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))) = (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋)))(-_g‘𝐹)(((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))))
162	76, 156, 160, 161	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) − (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))) = (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋)))(-_g‘𝐹)(((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))))
163	4, 5, 6, 7, 8, 15	tcphcphlem3 24750	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
164	13, 163	mpdan 686	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
165	164	recnd 11242	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
166	165	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑋) ∈ ℂ)
167	18	absvalsqd 15389	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌))↑2) = ((𝑋 , 𝑌) · (∗‘(𝑋 , 𝑌))))
168	66	oveq2d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑋 , 𝑌) · (∗‘(𝑋 , 𝑌))) = ((𝑋 , 𝑌) · (𝑌 , 𝑋)))
169	167, 168	eqtrd 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌))↑2) = ((𝑋 , 𝑌) · (𝑌 , 𝑋)))
170	18	abscld 15383	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
171	170	resqcld 14090	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌))↑2) ∈ ℝ)
172	169, 171	eqeltrrd 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ∈ ℝ)
173	172	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ∈ ℝ)
174	173, 71, 37	redivcld 12042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) ∈ ℝ)
175	41, 174	eqeltrrd 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · 𝐶) ∈ ℝ)
176	175	recnd 11242	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · 𝐶) ∈ ℂ)
177	76, 11	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 𝐾 ⊆ ℂ)
178	177, 154	sseldd 3984	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((∗‘𝐶) · (𝑌 , 𝑋)) ∈ ℂ)
179	166, 176, 178	pnpcan2d 11609	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋))) − (((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))) = ((𝑋 , 𝑋) − ((𝑋 , 𝑌) · 𝐶)))
180	162, 179	eqtr3d 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑋) + ((∗‘𝐶) · (𝑌 , 𝑋)))(-_g‘𝐹)(((𝑋 , 𝑌) · 𝐶) + ((∗‘𝐶) · (𝑌 , 𝑋)))) = ((𝑋 , 𝑋) − ((𝑋 , 𝑌) · 𝐶)))
181	103, 150, 180	3eqtrd 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌)) , (𝑋(-_g‘𝑊)((∗‘𝐶)( ·_𝑠 ‘𝑊)𝑌))) = ((𝑋 , 𝑋) − ((𝑋 , 𝑌) · 𝐶)))
182	99, 181	breqtrd 5175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 0 ≤ ((𝑋 , 𝑋) − ((𝑋 , 𝑌) · 𝐶)))
183	164	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (𝑋 , 𝑋) ∈ ℝ)
184	183, 175	subge0d 11804	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (0 ≤ ((𝑋 , 𝑋) − ((𝑋 , 𝑌) · 𝐶)) ↔ ((𝑋 , 𝑌) · 𝐶) ≤ (𝑋 , 𝑋)))
185	182, 184	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · 𝐶) ≤ (𝑋 , 𝑋))
186	41, 185	eqbrtrd 5171	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → (((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) ≤ (𝑋 , 𝑋))
187		oveq12 7418	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
188	187	anidms 568	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
189	188	breq2d 5161	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
190	189, 46, 14	rspcdva 3614	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
191	190	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 0 ≤ (𝑌 , 𝑌))
192	71, 191, 37	ne0gt0d 11351	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → 0 < (𝑌 , 𝑌))
193		ledivmul2 12093	. . . . . . 7 ⊢ ((((𝑋 , 𝑌) · (𝑌 , 𝑋)) ∈ ℝ ∧ (𝑋 , 𝑋) ∈ ℝ ∧ ((𝑌 , 𝑌) ∈ ℝ ∧ 0 < (𝑌 , 𝑌))) → ((((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) ≤ (𝑋 , 𝑋) ↔ ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌))))
194	173, 183, 71, 192, 193	syl112anc 1375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((((𝑋 , 𝑌) · (𝑌 , 𝑋)) / (𝑌 , 𝑌)) ≤ (𝑋 , 𝑋) ↔ ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌))))
195	186, 194	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ (0_g‘𝑊)) → ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
196	6, 15, 5, 31, 32	ip0r 21190	. . . . . . . . . 10 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉) → (𝑋 , (0_g‘𝑊)) = (0_g‘𝐹))
197	7, 13, 196	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 , (0_g‘𝑊)) = (0_g‘𝐹))
198	197, 29	eqtr4d 2776	. . . . . . . 8 ⊢ (𝜑 → (𝑋 , (0_g‘𝑊)) = 0)
199	198	oveq1d 7424	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , (0_g‘𝑊)) · (𝑌 , 𝑋)) = (0 · (𝑌 , 𝑋)))
200	22	mul02d 11412	. . . . . . 7 ⊢ (𝜑 → (0 · (𝑌 , 𝑋)) = 0)
201	199, 200	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → ((𝑋 , (0_g‘𝑊)) · (𝑌 , 𝑋)) = 0)
202		oveq12 7418	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
203	202	anidms 568	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
204	203	breq2d 5161	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
205	204, 46, 13	rspcdva 3614	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
206	164, 25, 205, 190	mulge0d 11791	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
207	201, 206	eqbrtrd 5171	. . . . 5 ⊢ (𝜑 → ((𝑋 , (0_g‘𝑊)) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
208	3, 195, 207	pm2.61ne 3028	. . . 4 ⊢ (𝜑 → ((𝑋 , 𝑌) · (𝑌 , 𝑋)) ≤ ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
209	164, 205	resqrtcld 15364	. . . . . . 7 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
210	209	recnd 11242	. . . . . 6 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
211	25, 190	resqrtcld 15364	. . . . . . 7 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
212	211	recnd 11242	. . . . . 6 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
213	210, 212	sqmuld 14123	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))↑2) = (((√‘(𝑋 , 𝑋))↑2) · ((√‘(𝑌 , 𝑌))↑2)))
214	165	sqsqrtd 15386	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
215	26	sqsqrtd 15386	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
216	214, 215	oveq12d 7427	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) · ((√‘(𝑌 , 𝑌))↑2)) = ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
217	213, 216	eqtrd 2773	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))↑2) = ((𝑋 , 𝑋) · (𝑌 , 𝑌)))
218	208, 169, 217	3brtr4d 5181	. . 3 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌))↑2) ≤ (((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))↑2))
219	209, 211	remulcld 11244	. . . 4 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
220	18	absge0d 15391	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝑋 , 𝑌)))
221	164, 205	sqrtge0d 15367	. . . . 5 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
222	25, 190	sqrtge0d 15367	. . . . 5 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
223	209, 211, 221, 222	mulge0d 11791	. . . 4 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
224	170, 219, 220, 223	le2sqd 14220	. . 3 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ ((abs‘(𝑋 , 𝑌))↑2) ≤ (((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))↑2)))
225	218, 224	mpbird 257	. 2 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
226		lmodgrp 20478	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
227	49, 226	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Grp)
228		ipcau2.n	. . . . 5 ⊢ 𝑁 = (norm‘𝐺)
229	4, 228, 5, 15	tcphnmval 24746	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) = (√‘(𝑋 , 𝑋)))
230	227, 13, 229	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑁‘𝑋) = (√‘(𝑋 , 𝑋)))
231	4, 228, 5, 15	tcphnmval 24746	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → (𝑁‘𝑌) = (√‘(𝑌 , 𝑌)))
232	227, 14, 231	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑁‘𝑌) = (√‘(𝑌 , 𝑌)))
233	230, 232	oveq12d 7427	. 2 ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
234	225, 233	breqtrrd 5177	1 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ⊆ wss 3949 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 2c2 12267 ↑cexp 14027 ∗ccj 15043 √csqrt 15180 abscabs 15181 Basecbs 17144 ↾_s cress 17173 +_gcplusg 17197 ._rcmulr 17198 *_𝑟cstv 17199 Scalarcsca 17200 ·_𝑠 cvsca 17201 ·_𝑖cip 17202 0_gc0g 17385 Grpcgrp 18819 -_gcsg 18821 DivRingcdr 20357 LModclmod 20471 LVecclvec 20713 ℂ_fldccnfld 20944 PreHilcphl 21177 normcnm 24085 ℂModcclm 24578 toℂPreHilctcph 24684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-ghm 19090 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-staf 20453 df-srng 20454 df-lmod 20473 df-lmhm 20633 df-lvec 20714 df-sra 20785 df-rgmod 20786 df-cnfld 20945 df-phl 21179 df-nm 24091 df-tng 24093 df-clm 24579 df-tcph 24686

This theorem is referenced by: tcphcphlem1 24752 ipcau 24755

W3C validator