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Theorem tcphcphlem1 25207

Description: Lemma for tcphcph 25209: 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 21579	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
3		lmodgrp 20762	. . . . . . 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 18984	. . . . . 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 25205	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
16	10, 15	mpdan 685	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ)
17	11, 7, 12, 1, 13, 14	tcphcphlem3 25205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ)
18	5, 17	mpdan 685	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ)
19	11, 7, 12, 1, 13, 14	tcphcphlem3 25205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ)
20	6, 19	mpdan 685	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ)
21	18, 20	readdcld 11275	. . . . 5 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ)
22	11, 7, 12, 1, 13	phclm 25204	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
23		tcphcph.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐹)
24	12, 23	clmsscn 25050	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
26	12, 14, 7, 23	ipcl 21582	. . . . . . . . 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 3977	. . . . . . 7 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
29	12, 14, 7, 23	ipcl 21582	. . . . . . . . 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 3977	. . . . . . 7 ⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ)
32	28, 31	addcld 11265	. . . . . 6 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ)
33	32	abscld 15419	. . . . 5 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ)
34	21, 33	readdcld 11275	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ)
35	18	recnd 11274	. . . . . 6 ⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ)
36		2re 12319	. . . . . . . 8 ⊢ 2 ∈ ℝ
37		oveq12 7428	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋))
38	37	anidms 565	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋))
39	38	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋)))
40		tcphcph.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥))
41	40	ralrimiva 3135	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥))
42	39, 41, 5	rspcdva 3607	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋))
43	18, 42	resqrtcld 15400	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ)
44		oveq12 7428	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌))
45	44	anidms 565	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌))
46	45	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌)))
47	46, 41, 6	rspcdva 3607	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌))
48	20, 47	resqrtcld 15400	. . . . . . . . 9 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ)
49	43, 48	remulcld 11276	. . . . . . . 8 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ)
50		remulcl 11225	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
51	36, 49, 50	sylancr 585	. . . . . . 7 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℝ)
52	51	recnd 11274	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) ∈ ℂ)
53	20	recnd 11274	. . . . . 6 ⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ)
54	35, 52, 53	add32d 11473	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
55	21, 51	readdcld 11275	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ)
56	54, 55	eqeltrd 2825	. . . 4 ⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ)
57		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
58	57	anidms 565	. . . . . . . . . 10 ⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
59	58	breq2d 5161	. . . . . . . . 9 ⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))))
60	59, 41, 10	rspcdva 3607	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
61	16, 60	absidd 15405	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
62	12	clmadd 25045	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → + = (+_g‘𝐹))
63	22, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+_g‘𝐹))
64	63	oveqd 7436	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+_g‘𝐹)(𝑌 , 𝑌)))
65	63	oveqd 7436	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+_g‘𝐹)(𝑌 , 𝑋)))
66	64, 65	oveq12d 7437	. . . . . . . . 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 21582	. . . . . . . . . . . 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 21582	. . . . . . . . . . . 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 25055	. . . . . . . . . . 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 25055	. . . . . . . . . . 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 25051	. . . . . . . . . 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 2725	. . . . . . . . . 10 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
78		eqid 2725	. . . . . . . . . 10 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
79	12, 14, 7, 8, 77, 78, 1, 5, 6, 5, 6	ip2subdi 21593	. . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))
81	80	fveq2d 6900	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
82	61, 81	eqtr3d 2767	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
83	25, 72	sseldd 3977	. . . . . . 7 ⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ)
84	83, 32	abs2dif2d 15441	. . . . . 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 5171	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
86	18, 20, 42, 47	addge0d 11822	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
87	21, 86	absidd 15405	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌)))
88	87	oveq1d 7434	. . . . 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 5175	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))))
90	28	abscld 15419	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ)
91		remulcl 11225	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
92	36, 90, 91	sylancr 585	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ∈ ℝ)
93	28, 31	abstrid 15439	. . . . . . . 8 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
94	90	recnd 11274	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ)
95	94	2timesd 12488	. . . . . . . . 9 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))))
96	28	abscjd 15433	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌)))
97	12	clmcj 25047	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ ℂMod → ∗ = (*_𝑟‘𝐹))
98	22, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*_𝑟‘𝐹))
99	98	fveq1d 6898	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*_𝑟‘𝐹)‘(𝑋 , 𝑌)))
100		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (_𝑟‘𝐹) = (_𝑟‘𝐹)
101	12, 14, 7, 100	ipcj 21583	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋))
104	103	fveq2d 6900	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋)))
105	96, 104	eqtr3d 2767	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋)))
106	105	oveq2d 7435	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
107	95, 106	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋))))
108	93, 107	breqtrrd 5177	. . . . . . 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 2725	. . . . . . . . . 10 ⊢ (norm‘𝐺) = (norm‘𝐺)
111		eqid 2725	. . . . . . . . . 10 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌))
112	11, 7, 12, 1, 13, 14, 109, 40, 23, 110, 111, 5, 6	ipcau2 25206	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)))
113	11, 110, 7, 14	tcphnmval 25201	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
114	4, 5, 113	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋)))
115	11, 110, 7, 14	tcphnmval 25201	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
116	4, 6, 115	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌)))
117	114, 116	oveq12d 7437	. . . . . . . . 9 ⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
118	112, 117	breqtrd 5175	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))
119	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
120		2pos 12348	. . . . . . . . . 10 ⊢ 0 < 2
121	120	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
122		lemul2 12100	. . . . . . . . 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 231	. . . . . . 7 ⊢ (𝜑 → (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
125	33, 92, 51, 108, 124	letrd 11403	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))
126	33, 51, 21, 125	leadd2dd 11861	. . . . 5 ⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
127	126, 54	breqtrrd 5177	. . . 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 11403	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
129	16	recnd 11274	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ)
130	129	sqsqrtd 15422	. . 3 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌)))
131	35	sqrtcld 15420	. . . . 5 ⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ)
132	48	recnd 11274	. . . . 5 ⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ)
133		binom2 14216	. . . . 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 582	. . . 4 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)))
135	35	sqsqrtd 15422	. . . . . 6 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋))
136	135	oveq1d 7434	. . . . 5 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))))
137	53	sqsqrtd 15422	. . . . 5 ⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌))
138	136, 137	oveq12d 7437	. . . 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 2765	. . 3 ⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)))
140	128, 130, 139	3brtr4d 5181	. 2 ⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))
141	16, 60	resqrtcld 15400	. . 3 ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ)
142	43, 48	readdcld 11275	. . 3 ⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ)
143	16, 60	sqrtge0d 15403	. . 3 ⊢ (𝜑 → 0 ≤ (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))))
144	18, 42	sqrtge0d 15403	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑋 , 𝑋)))
145	20, 47	sqrtge0d 15403	. . . 4 ⊢ (𝜑 → 0 ≤ (√‘(𝑌 , 𝑌)))
146	43, 48, 144, 145	addge0d 11822	. . 3 ⊢ (𝜑 → 0 ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))
147	141, 142, 143, 146	le2sqd 14255	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 + caddc 11143 · cmul 11145 < clt 11280 ≤ cle 11281 − cmin 11476 / cdiv 11903 2c2 12300 ↑cexp 14062 ∗ccj 15079 √csqrt 15216 abscabs 15217 Basecbs 17183 ↾_s cress 17212 +_gcplusg 17236 *_𝑟cstv 17238 Scalarcsca 17239 ·_𝑖cip 17241 Grpcgrp 18898 -_gcsg 18900 LModclmod 20755 ℂ_fldccnfld 21296 PreHilcphl 21573 normcnm 24529 ℂModcclm 25033 toℂPreHilctcph 25139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-ghm 19176 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-drng 20638 df-staf 20737 df-srng 20738 df-lmod 20757 df-lmhm 20919 df-lvec 21000 df-sra 21070 df-rgmod 21071 df-cnfld 21297 df-phl 21575 df-nm 24535 df-tng 24537 df-clm 25034 df-tcph 25141

This theorem is referenced by: tcphcph 25209

W3C validator