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Theorem tcphcphlem1 25284

Description: Lemma for tcphcph 25286: 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 21669	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
3		lmodgrp 20921	. . . . . . 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 19052	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
10	4, 5, 6, 9	syl3anc 1389	. . . . 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 25282	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
16	10, 15	mpdan 697	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
17	11, 7, 12, 1, 13, 14	tcphcphlem3 25282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
18	5, 17	mpdan 697	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
19	11, 7, 12, 1, 13, 14	tcphcphlem3 25282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
20	6, 19	mpdan 697	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
21	18, 20	readdcld 11204	. . . . 5 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ)
22	11, 7, 12, 1, 13	phclm 25281	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
23		tcphcph.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
24	12, 23	clmsscn 25128	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
26	12, 14, 7, 23	ipcl 21672	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾)
27	1, 5, 6, 26	syl3anc 1389	. . . . . . . 8 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾)
28	25, 27	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
29	12, 14, 7, 23	ipcl 21672	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾)
30	1, 6, 5, 29	syl3anc 1389	. . . . . . . 8 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾)
31	25, 30	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
32	28, 31	addcld 11194	. . . . . 6 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ)
33	32	abscld 15456	. . . . 5 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ)
34	21, 33	readdcld 11204	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ)
35	18	recnd 11203	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
36		2re 12285	. . . . . . . 8 ⊢ 2 ∈ ℝ
37		oveq12 7399	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
38	37	anidms 574	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
39	38	breq2d 5109	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
40		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
41	40	ralrimiva 3153	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
42	39, 41, 5	rspcdva 3581	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
43	18, 42	resqrtcld 15435	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
44		oveq12 7399	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
45	44	anidms 574	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
46	45	breq2d 5109	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
47	46, 41, 6	rspcdva 3581	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
48	20, 47	resqrtcld 15435	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
49	43, 48	remulcld 11205	. . . . . . . 8 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
50		remulcl 11151	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
51	36, 49, 50	sylancr 596	. . . . . . 7 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
52	51	recnd 11203	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℂ)
53	20	recnd 11203	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
54	35, 52, 53	add32d 11404	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
55	21, 51	readdcld 11204	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ)
56	54, 55	eqeltrd 2861	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ)
57		oveq12 7399	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
58	57	anidms 574	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
59	58	breq2d 5109	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))))
60	59, 41, 10	rspcdva 3581	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
61	16, 60	absidd 15440	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
62	12	clmadd 25123	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → + = (+_g‘𝐹))
63	22, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+_g‘𝐹))
64	63	oveqd 7407	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌)))
65	63	oveqd 7407	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋)))
66	64, 65	oveq12d 7408	. . . . . . . . 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 21672	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾)
68	1, 5, 5, 67	syl3anc 1389	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ 𝐾)
69	12, 14, 7, 23	ipcl 21672	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾)
70	1, 6, 6, 69	syl3anc 1389	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾)
71	12, 23	clmacl 25133	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ (𝑌 , 𝑌) ∈ 𝐾) → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
72	22, 68, 70, 71	syl3anc 1389	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
73	12, 23	clmacl 25133	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
74	22, 27, 30, 73	syl3anc 1389	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
75	12, 23	clmsub 25129	. . . . . . . . . 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 1389	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))))
77		eqid 2761	. . . . . . . . . 10 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
78		eqid 2761	. . . . . . . . . 10 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
79	12, 14, 7, 8, 77, 78, 1, 5, 6, 5, 6	ip2subdi 21683	. . . . . . . . 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 2807	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))
81	80	fveq2d 6865	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
82	61, 81	eqtr3d 2798	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
83	25, 72	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ)
84	83, 32	abs2dif2d 15478	. . . . . 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 5119	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
86	18, 20, 42, 47	addge0d 11756	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
87	21, 86	absidd 15440	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
88	87	oveq1d 7405	. . . . 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 5123	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
90	28	abscld 15456	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
91		remulcl 11151	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
92	36, 90, 91	sylancr 596	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
93	28, 31	abstrid 15476	. . . . . . . 8 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
94	90	recnd 11203	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ)
95	94	2timesd 12457	. . . . . . . . 9 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))))
96	28	abscjd 15470	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌)))
97	12	clmcj 25125	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
98	22, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
99	98	fveq1d 6863	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)))
100		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
101	12, 14, 7, 100	ipcj 21673	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
102	1, 5, 6, 101	syl3anc 1389	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
103	99, 102	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
104	103	fveq2d 6865	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋)))
105	96, 104	eqtr3d 2798	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋)))
106	105	oveq2d 7406	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
107	95, 106	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
108	93, 107	breqtrrd 5125	. . . . . . 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 2761	. . . . . . . . . 10 ⊢ (norm‘𝐺) = (norm‘𝐺)
111		eqid 2761	. . . . . . . . . 10 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
112	11, 7, 12, 1, 13, 14, 109, 40, 23, 110, 111, 5, 6	ipcau2 25283	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)))
113	11, 110, 7, 14	tcphnmval 25278	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
114	4, 5, 113	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
115	11, 110, 7, 14	tcphnmval 25278	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
116	4, 6, 115	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
117	114, 116	oveq12d 7408	. . . . . . . . 9 ⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
118	112, 117	breqtrd 5123	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
119	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
120		2pos 12315	. . . . . . . . . 10 ⊢ 0 < 2
121	120	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
122		lemul2 12037	. . . . . . . . 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 1392	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
124	118, 123	mpbid 234	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
125	33, 92, 51, 108, 124	letrd 11333	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
126	33, 51, 21, 125	leadd2dd 11795	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
127	126, 54	breqtrrd 5125	. . . 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 11333	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
129	16	recnd 11203	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ)
130	129	sqsqrtd 15459	. . 3 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
131	35	sqrtcld 15457	. . . . 5 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
132	48	recnd 11203	. . . . 5 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
133		binom2 14223	. . . . 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 593	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
135	35	sqsqrtd 15459	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
136	135	oveq1d 7405	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
137	53	sqsqrtd 15459	. . . . 5 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
138	136, 137	oveq12d 7408	. . . 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 2796	. . 3 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
140	128, 130, 139	3brtr4d 5129	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))
141	16, 60	resqrtcld 15435	. . 3 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ)
142	43, 48	readdcld 11204	. . 3 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ)
143	16, 60	sqrtge0d 15438	. . 3 ⊢ (𝜑 → 0 ≤ (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))))
144	18, 42	sqrtge0d 15438	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
145	20, 47	sqrtge0d 15438	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
146	43, 48, 144, 145	addge0d 11756	. . 3 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
147	141, 142, 143, 146	le2sqd 14263	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ↔ ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2)))
148	140, 147	mpbird 259	1 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ⊆ wss 3902 class class class wbr 5097 ‘cfv 6515 (class class class)co 7390 ℂcc 11064 ℝcr 11065 0cc0 11066 + caddc 11069 · cmul 11071 < clt 11209 ≤ cle 11210 − cmin 11407 / cdiv 11837 2c2 12265 ↑cexp 14067 ∗ccj 15113 √csqrt 15250 abscabs 15251 Basecbs 17235 ↾_s cress 17256 +_gcplusg 17276 *_𝑟cstv 17278 Scalarcsca 17279 ·_𝑖cip 17281 Grpcgrp 18965 -_gcsg 18967 LModclmod 20914 ℂ_fldccnfld 21411 PreHilcphl 21663 normcnm 24623 ℂModcclm 25111 toℂPreHilctcph 25216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145 ax-mulf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9381 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-rp 12987 df-fz 13506 df-seq 14008 df-exp 14068 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-0g 17460 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-grp 18968 df-minusg 18969 df-sbg 18970 df-subg 19155 df-ghm 19244 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-rhm 20507 df-subrng 20582 df-subrg 20606 df-drng 20767 df-staf 20875 df-srng 20876 df-lmod 20916 df-lmhm 21076 df-lvec 21157 df-sra 21227 df-rgmod 21228 df-cnfld 21412 df-phl 21665 df-nm 24629 df-tng 24631 df-clm 25112 df-tcph 25218

This theorem is referenced by: tcphcph 25286

W3C validator