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Theorem tcphcphlem1 23839

Description: Lemma for tcphcph 23841: 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 20319	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
3		lmodgrp 19634	. . . . . . 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 18171	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
10	4, 5, 6, 9	syl3anc 1368	. . . . 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 23837	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
16	10, 15	mpdan 686	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
17	11, 7, 12, 1, 13, 14	tcphcphlem3 23837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
18	5, 17	mpdan 686	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
19	11, 7, 12, 1, 13, 14	tcphcphlem3 23837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
20	6, 19	mpdan 686	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
21	18, 20	readdcld 10659	. . . . 5 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ)
22	11, 7, 12, 1, 13	phclm 23836	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
23		tcphcph.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
24	12, 23	clmsscn 23684	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
26	12, 14, 7, 23	ipcl 20322	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾)
27	1, 5, 6, 26	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾)
28	25, 27	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
29	12, 14, 7, 23	ipcl 20322	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾)
30	1, 6, 5, 29	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾)
31	25, 30	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
32	28, 31	addcld 10649	. . . . . 6 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ)
33	32	abscld 14788	. . . . 5 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ)
34	21, 33	readdcld 10659	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ)
35	18	recnd 10658	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
36		2re 11699	. . . . . . . 8 ⊢ 2 ∈ ℝ
37		oveq12 7144	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
38	37	anidms 570	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
39	38	breq2d 5042	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
40		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
41	40	ralrimiva 3149	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
42	39, 41, 5	rspcdva 3573	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
43	18, 42	resqrtcld 14769	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
44		oveq12 7144	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
45	44	anidms 570	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
46	45	breq2d 5042	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
47	46, 41, 6	rspcdva 3573	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
48	20, 47	resqrtcld 14769	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
49	43, 48	remulcld 10660	. . . . . . . 8 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
50		remulcl 10611	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
51	36, 49, 50	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
52	51	recnd 10658	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℂ)
53	20	recnd 10658	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
54	35, 52, 53	add32d 10856	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
55	21, 51	readdcld 10659	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ)
56	54, 55	eqeltrd 2890	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ)
57		oveq12 7144	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
58	57	anidms 570	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
59	58	breq2d 5042	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))))
60	59, 41, 10	rspcdva 3573	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
61	16, 60	absidd 14774	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
62	12	clmadd 23679	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → + = (+_g‘𝐹))
63	22, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+_g‘𝐹))
64	63	oveqd 7152	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌)))
65	63	oveqd 7152	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋)))
66	64, 65	oveq12d 7153	. . . . . . . . 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 20322	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾)
68	1, 5, 5, 67	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ 𝐾)
69	12, 14, 7, 23	ipcl 20322	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾)
70	1, 6, 6, 69	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾)
71	12, 23	clmacl 23689	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ (𝑌 , 𝑌) ∈ 𝐾) → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
72	22, 68, 70, 71	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾)
73	12, 23	clmacl 23689	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
74	22, 27, 30, 73	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾)
75	12, 23	clmsub 23685	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-_g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))))
77		eqid 2798	. . . . . . . . . 10 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
78		eqid 2798	. . . . . . . . . 10 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
79	12, 14, 7, 8, 77, 78, 1, 5, 6, 5, 6	ip2subdi 20333	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))
81	80	fveq2d 6649	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
82	61, 81	eqtr3d 2835	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
83	25, 72	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ)
84	83, 32	abs2dif2d 14810	. . . . . 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 5052	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
86	18, 20, 42, 47	addge0d 11205	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
87	21, 86	absidd 14774	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
88	87	oveq1d 7150	. . . . 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 5056	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
90	28	abscld 14788	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
91		remulcl 10611	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
92	36, 90, 91	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
93	28, 31	abstrid 14808	. . . . . . . 8 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
94	90	recnd 10658	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ)
95	94	2timesd 11868	. . . . . . . . 9 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))))
96	28	abscjd 14802	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌)))
97	12	clmcj 23681	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
98	22, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
99	98	fveq1d 6647	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)))
100		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
101	12, 14, 7, 100	ipcj 20323	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
102	1, 5, 6, 101	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
103	99, 102	eqtrd 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
104	103	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋)))
105	96, 104	eqtr3d 2835	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋)))
106	105	oveq2d 7151	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
107	95, 106	eqtrd 2833	. . . . . . . 8 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
108	93, 107	breqtrrd 5058	. . . . . . 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 2798	. . . . . . . . . 10 ⊢ (norm‘𝐺) = (norm‘𝐺)
111		eqid 2798	. . . . . . . . . 10 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
112	11, 7, 12, 1, 13, 14, 109, 40, 23, 110, 111, 5, 6	ipcau2 23838	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)))
113	11, 110, 7, 14	tcphnmval 23833	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
114	4, 5, 113	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
115	11, 110, 7, 14	tcphnmval 23833	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
116	4, 6, 115	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
117	114, 116	oveq12d 7153	. . . . . . . . 9 ⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
118	112, 117	breqtrd 5056	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
119	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
120		2pos 11728	. . . . . . . . . 10 ⊢ 0 < 2
121	120	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
122		lemul2 11482	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
124	118, 123	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
125	33, 92, 51, 108, 124	letrd 10786	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
126	33, 51, 21, 125	leadd2dd 11244	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
127	126, 54	breqtrrd 5058	. . . 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 10786	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
129	16	recnd 10658	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ)
130	129	sqsqrtd 14791	. . 3 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
131	35	sqrtcld 14789	. . . . 5 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
132	48	recnd 10658	. . . . 5 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
133		binom2 13575	. . . . 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 587	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
135	35	sqsqrtd 14791	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
136	135	oveq1d 7150	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
137	53	sqsqrtd 14791	. . . . 5 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
138	136, 137	oveq12d 7153	. . . 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 2833	. . 3 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
140	128, 130, 139	3brtr4d 5062	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))
141	16, 60	resqrtcld 14769	. . 3 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ)
142	43, 48	readdcld 10659	. . 3 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ)
143	16, 60	sqrtge0d 14772	. . 3 ⊢ (𝜑 → 0 ≤ (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))))
144	18, 42	sqrtge0d 14772	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
145	20, 47	sqrtge0d 14772	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
146	43, 48, 144, 145	addge0d 11205	. . 3 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
147	141, 142, 143, 146	le2sqd 13616	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ↔ ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2)))
148	140, 147	mpbird 260	1 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ↑cexp 13425 ∗ccj 14447 √csqrt 14584 abscabs 14585 Basecbs 16475 ↾_s cress 16476 +_gcplusg 16557 *_𝑟cstv 16559 Scalarcsca 16560 ·_𝑖cip 16562 Grpcgrp 18095 -_gcsg 18097 LModclmod 19627 ℂ_fldccnfld 20091 PreHilcphl 20313 normcnm 23183 ℂModcclm 23667 toℂPreHilctcph 23772

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lmhm 19787 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-phl 20315 df-nm 23189 df-tng 23191 df-clm 23668 df-tcph 23774

This theorem is referenced by: tcphcph 23841

W3C validator