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Theorem tcphcphlem1 24399

Description: Lemma for tcphcph 24401: 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 20835	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
3		lmodgrp 20130	. . . . . . 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 18655	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
10	4, 5, 6, 9	syl3anc 1370	. . . . 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 24397	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
16	10, 15	mpdan 684	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
17	11, 7, 12, 1, 13, 14	tcphcphlem3 24397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
18	5, 17	mpdan 684	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
19	11, 7, 12, 1, 13, 14	tcphcphlem3 24397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
20	6, 19	mpdan 684	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
21	18, 20	readdcld 11004	. . . . 5 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ)
22	11, 7, 12, 1, 13	phclm 24396	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
23		tcphcph.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
24	12, 23	clmsscn 24242	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
26	12, 14, 7, 23	ipcl 20838	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾)
27	1, 5, 6, 26	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾)
28	25, 27	sseldd 3922	. . . . . . 7 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
29	12, 14, 7, 23	ipcl 20838	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾)
30	1, 6, 5, 29	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾)
31	25, 30	sseldd 3922	. . . . . . 7 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
32	28, 31	addcld 10994	. . . . . 6 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ)
33	32	abscld 15148	. . . . 5 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ)
34	21, 33	readdcld 11004	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ)
35	18	recnd 11003	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
36		2re 12047	. . . . . . . 8 ⊢ 2 ∈ ℝ
37		oveq12 7284	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
38	37	anidms 567	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
39	38	breq2d 5086	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
40		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
41	40	ralrimiva 3103	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
42	39, 41, 5	rspcdva 3562	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
43	18, 42	resqrtcld 15129	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
44		oveq12 7284	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
45	44	anidms 567	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
46	45	breq2d 5086	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
47	46, 41, 6	rspcdva 3562	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
48	20, 47	resqrtcld 15129	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
49	43, 48	remulcld 11005	. . . . . . . 8 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
50		remulcl 10956	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
51	36, 49, 50	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
52	51	recnd 11003	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℂ)
53	20	recnd 11003	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
54	35, 52, 53	add32d 11202	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
55	21, 51	readdcld 11004	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ)
56	54, 55	eqeltrd 2839	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ)
57		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
58	57	anidms 567	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
59	58	breq2d 5086	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))))
60	59, 41, 10	rspcdva 3562	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
61	16, 60	absidd 15134	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
62	12	clmadd 24237	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → + = (+_g‘𝐹))
63	22, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+_g‘𝐹))
64	63	oveqd 7292	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌)))
65	63	oveqd 7292	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋)))
66	64, 65	oveq12d 7293	. . . . . . . . 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 20838	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾)
68	1, 5, 5, 67	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ 𝐾)
69	12, 14, 7, 23	ipcl 20838	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾)
70	1, 6, 6, 69	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾)
71	12, 23	clmacl 24247	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ (𝑌 , 𝑌) ∈ 𝐾) → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
72	22, 68, 70, 71	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
73	12, 23	clmacl 24247	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
74	22, 27, 30, 73	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
75	12, 23	clmsub 24243	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))))
77		eqid 2738	. . . . . . . . . 10 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
78		eqid 2738	. . . . . . . . . 10 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
79	12, 14, 7, 8, 77, 78, 1, 5, 6, 5, 6	ip2subdi 20849	. . . . . . . . 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 2789	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))
81	80	fveq2d 6778	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
82	61, 81	eqtr3d 2780	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
83	25, 72	sseldd 3922	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ)
84	83, 32	abs2dif2d 15170	. . . . . 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 5096	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
86	18, 20, 42, 47	addge0d 11551	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
87	21, 86	absidd 15134	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
88	87	oveq1d 7290	. . . . 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 5100	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
90	28	abscld 15148	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
91		remulcl 10956	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
92	36, 90, 91	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
93	28, 31	abstrid 15168	. . . . . . . 8 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
94	90	recnd 11003	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ)
95	94	2timesd 12216	. . . . . . . . 9 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))))
96	28	abscjd 15162	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌)))
97	12	clmcj 24239	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
98	22, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
99	98	fveq1d 6776	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)))
100		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
101	12, 14, 7, 100	ipcj 20839	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
102	1, 5, 6, 101	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
103	99, 102	eqtrd 2778	. . . . . . . . . . . 12 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
104	103	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋)))
105	96, 104	eqtr3d 2780	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋)))
106	105	oveq2d 7291	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
107	95, 106	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
108	93, 107	breqtrrd 5102	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (norm‘𝐺) = (norm‘𝐺)
111		eqid 2738	. . . . . . . . . 10 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
112	11, 7, 12, 1, 13, 14, 109, 40, 23, 110, 111, 5, 6	ipcau2 24398	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)))
113	11, 110, 7, 14	tcphnmval 24393	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
114	4, 5, 113	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
115	11, 110, 7, 14	tcphnmval 24393	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
116	4, 6, 115	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
117	114, 116	oveq12d 7293	. . . . . . . . 9 ⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
118	112, 117	breqtrd 5100	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
119	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
120		2pos 12076	. . . . . . . . . 10 ⊢ 0 < 2
121	120	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
122		lemul2 11828	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
124	118, 123	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
125	33, 92, 51, 108, 124	letrd 11132	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
126	33, 51, 21, 125	leadd2dd 11590	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
127	126, 54	breqtrrd 5102	. . . 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 11132	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
129	16	recnd 11003	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ)
130	129	sqsqrtd 15151	. . 3 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
131	35	sqrtcld 15149	. . . . 5 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
132	48	recnd 11003	. . . . 5 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
133		binom2 13933	. . . . 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 584	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
135	35	sqsqrtd 15151	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
136	135	oveq1d 7290	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
137	53	sqsqrtd 15151	. . . . 5 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
138	136, 137	oveq12d 7293	. . . 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 2778	. . 3 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
140	128, 130, 139	3brtr4d 5106	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))
141	16, 60	resqrtcld 15129	. . 3 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ)
142	43, 48	readdcld 11004	. . 3 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ)
143	16, 60	sqrtge0d 15132	. . 3 ⊢ (𝜑 → 0 ≤ (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))))
144	18, 42	sqrtge0d 15132	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
145	20, 47	sqrtge0d 15132	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
146	43, 48, 144, 145	addge0d 11551	. . 3 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
147	141, 142, 143, 146	le2sqd 13974	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ↔ ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2)))
148	140, 147	mpbird 256	1 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 ↑cexp 13782 ∗ccj 14807 √csqrt 14944 abscabs 14945 Basecbs 16912 ↾_s cress 16941 +_gcplusg 16962 *_𝑟cstv 16964 Scalarcsca 16965 ·_𝑖cip 16967 Grpcgrp 18577 -_gcsg 18579 LModclmod 20123 ℂ_fldccnfld 20597 PreHilcphl 20829 normcnm 23732 ℂModcclm 24225 toℂPreHilctcph 24331

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lmhm 20284 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-phl 20831 df-nm 23738 df-tng 23740 df-clm 24226 df-tcph 24333

This theorem is referenced by: tcphcph 24401

W3C validator