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Theorem tcphcphlem1 25212

Description: Lemma for tcphcph 25214: the triangle inequality. (Contributed by Mario Carneiro, 8-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)
tcphcph.m	⊢ − = (-_g‘𝑊)
tcphcphlem1.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
tcphcphlem1.4	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
tcphcphlem1	⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

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

**Proof of Theorem tcphcphlem1**
Step	Hyp	Ref	Expression
1		tcphcph.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ PreHil)
2		phllmod 21620	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
3		lmodgrp 20853	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
4	1, 2, 3	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
5		tcphcphlem1.3	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		tcphcphlem1.4	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		tcphcph.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
8		tcphcph.m	. . . . . . 7 ⊢ − = (-_g‘𝑊)
9	7, 8	grpsubcl 18987	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
10	4, 5, 6, 9	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
11		tcphval.n	. . . . . 6 ⊢ 𝐺 = (toℂPreHil‘𝑊)
12		tcphcph.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
13		tcphcph.2	. . . . . 6 ⊢ (𝜑 → 𝐹 = (ℂ_fld ↾_s 𝐾))
14		tcphcph.h	. . . . . 6 ⊢ , = (·_𝑖‘𝑊)
15	11, 7, 12, 1, 13, 14	tcphcphlem3 25210	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
16	10, 15	mpdan 688	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
17	11, 7, 12, 1, 13, 14	tcphcphlem3 25210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
18	5, 17	mpdan 688	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
19	11, 7, 12, 1, 13, 14	tcphcphlem3 25210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
20	6, 19	mpdan 688	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
21	18, 20	readdcld 11165	. . . . 5 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ)
22	11, 7, 12, 1, 13	phclm 25209	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
23		tcphcph.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
24	12, 23	clmsscn 25056	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
26	12, 14, 7, 23	ipcl 21623	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾)
27	1, 5, 6, 26	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾)
28	25, 27	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
29	12, 14, 7, 23	ipcl 21623	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾)
30	1, 6, 5, 29	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾)
31	25, 30	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
32	28, 31	addcld 11155	. . . . . 6 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ)
33	32	abscld 15392	. . . . 5 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ)
34	21, 33	readdcld 11165	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ)
35	18	recnd 11164	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
36		2re 12246	. . . . . . . 8 ⊢ 2 ∈ ℝ
37		oveq12 7369	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
38	37	anidms 566	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
39	38	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
40		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
41	40	ralrimiva 3130	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
42	39, 41, 5	rspcdva 3566	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
43	18, 42	resqrtcld 15371	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
44		oveq12 7369	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
45	44	anidms 566	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
46	45	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
47	46, 41, 6	rspcdva 3566	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
48	20, 47	resqrtcld 15371	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
49	43, 48	remulcld 11166	. . . . . . . 8 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
50		remulcl 11114	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
51	36, 49, 50	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
52	51	recnd 11164	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℂ)
53	20	recnd 11164	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
54	35, 52, 53	add32d 11365	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
55	21, 51	readdcld 11165	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ)
56	54, 55	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ)
57		oveq12 7369	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
58	57	anidms 566	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
59	58	breq2d 5098	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))))
60	59, 41, 10	rspcdva 3566	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
61	16, 60	absidd 15376	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
62	12	clmadd 25051	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → + = (+_g‘𝐹))
63	22, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+_g‘𝐹))
64	63	oveqd 7377	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌)))
65	63	oveqd 7377	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋)))
66	64, 65	oveq12d 7378	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋))))
67	12, 14, 7, 23	ipcl 21623	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾)
68	1, 5, 5, 67	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ 𝐾)
69	12, 14, 7, 23	ipcl 21623	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾)
70	1, 6, 6, 69	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾)
71	12, 23	clmacl 25061	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ (𝑌 , 𝑌) ∈ 𝐾) → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
72	22, 68, 70, 71	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
73	12, 23	clmacl 25061	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
74	22, 27, 30, 73	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
75	12, 23	clmsub 25057	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂMod ∧ ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾 ∧ ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾) → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))))
76	22, 72, 74, 75	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))))
77		eqid 2737	. . . . . . . . . 10 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
78		eqid 2737	. . . . . . . . . 10 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
79	12, 14, 7, 8, 77, 78, 1, 5, 6, 5, 6	ip2subdi 21634	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋))))
80	66, 76, 79	3eqtr4rd 2783	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))
81	80	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
82	61, 81	eqtr3d 2774	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
83	25, 72	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ)
84	83, 32	abs2dif2d 15414	. . . . . 6 ⊢ (𝜑 → (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
85	82, 84	eqbrtrd 5108	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
86	18, 20, 42, 47	addge0d 11717	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
87	21, 86	absidd 15376	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
88	87	oveq1d 7375	. . . . 5 ⊢ (𝜑 → ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
89	85, 88	breqtrd 5112	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
90	28	abscld 15392	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
91		remulcl 11114	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
92	36, 90, 91	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
93	28, 31	abstrid 15412	. . . . . . . 8 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
94	90	recnd 11164	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ)
95	94	2timesd 12411	. . . . . . . . 9 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))))
96	28	abscjd 15406	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌)))
97	12	clmcj 25053	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
98	22, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
99	98	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)))
100		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
101	12, 14, 7, 100	ipcj 21624	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
102	1, 5, 6, 101	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
103	99, 102	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
104	103	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋)))
105	96, 104	eqtr3d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋)))
106	105	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
107	95, 106	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
108	93, 107	breqtrrd 5114	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · (abs‘(𝑋 , 𝑌))))
109		tcphcph.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)
110		eqid 2737	. . . . . . . . . 10 ⊢ (norm‘𝐺) = (norm‘𝐺)
111		eqid 2737	. . . . . . . . . 10 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
112	11, 7, 12, 1, 13, 14, 109, 40, 23, 110, 111, 5, 6	ipcau2 25211	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)))
113	11, 110, 7, 14	tcphnmval 25206	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
114	4, 5, 113	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
115	11, 110, 7, 14	tcphnmval 25206	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
116	4, 6, 115	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
117	114, 116	oveq12d 7378	. . . . . . . . 9 ⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
118	112, 117	breqtrd 5112	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
119	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
120		2pos 12275	. . . . . . . . . 10 ⊢ 0 < 2
121	120	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
122		lemul2 11999	. . . . . . . . 9 ⊢ (((abs‘(𝑋 , 𝑌)) ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
123	90, 49, 119, 121, 122	syl112anc 1377	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
124	118, 123	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
125	33, 92, 51, 108, 124	letrd 11294	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
126	33, 51, 21, 125	leadd2dd 11756	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
127	126, 54	breqtrrd 5114	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
128	16, 34, 56, 89, 127	letrd 11294	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
129	16	recnd 11164	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ)
130	129	sqsqrtd 15395	. . 3 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
131	35	sqrtcld 15393	. . . . 5 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
132	48	recnd 11164	. . . . 5 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
133		binom2 14170	. . . . 5 ⊢ (((√‘(𝑋 , 𝑋)) ∈ ℂ ∧ (√‘(𝑌 , 𝑌)) ∈ ℂ) → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
134	131, 132, 133	syl2anc 585	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
135	35	sqsqrtd 15395	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
136	135	oveq1d 7375	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
137	53	sqsqrtd 15395	. . . . 5 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
138	136, 137	oveq12d 7378	. . . 4 ⊢ (𝜑 → ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
139	134, 138	eqtrd 2772	. . 3 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
140	128, 130, 139	3brtr4d 5118	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))
141	16, 60	resqrtcld 15371	. . 3 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ)
142	43, 48	readdcld 11165	. . 3 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ)
143	16, 60	sqrtge0d 15374	. . 3 ⊢ (𝜑 → 0 ≤ (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))))
144	18, 42	sqrtge0d 15374	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
145	20, 47	sqrtge0d 15374	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
146	43, 48, 144, 145	addge0d 11717	. . 3 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
147	141, 142, 143, 146	le2sqd 14210	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ↔ ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2)))
148	140, 147	mpbird 257	1 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 2c2 12227 ↑cexp 14014 ∗ccj 15049 √csqrt 15186 abscabs 15187 Basecbs 17170 ↾_s cress 17191 +_gcplusg 17211 *_𝑟cstv 17213 Scalarcsca 17214 ·_𝑖cip 17216 Grpcgrp 18900 -_gcsg 18902 LModclmod 20846 ℂ_fldccnfld 21344 PreHilcphl 21614 normcnm 24551 ℂModcclm 25039 toℂPreHilctcph 25144

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-ghm 19179 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20514 df-subrg 20538 df-drng 20699 df-staf 20807 df-srng 20808 df-lmod 20848 df-lmhm 21009 df-lvec 21090 df-sra 21160 df-rgmod 21161 df-cnfld 21345 df-phl 21616 df-nm 24557 df-tng 24559 df-clm 25040 df-tcph 25146

This theorem is referenced by: tcphcph 25214

W3C validator