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Theorem imsmetlem 30719

Description: Lemma for imsmet 30720. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
imsmetlem.1	⊢ 𝑋 = (BaseSet‘𝑈)
imsmetlem.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
imsmetlem.7	⊢ 𝑀 = (inv‘𝐺)
imsmetlem.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
imsmetlem.5	⊢ 𝑍 = (0_vec‘𝑈)
imsmetlem.6	⊢ 𝑁 = (norm_CV‘𝑈)
imsmetlem.8	⊢ 𝐷 = (IndMet‘𝑈)
imsmetlem.9	⊢ 𝑈 ∈ NrmCVec

**Assertion**
Ref	Expression
imsmetlem	⊢ 𝐷 ∈ (Met‘𝑋)

Proof of Theorem imsmetlem Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imsmetlem.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
2	1	fvexi 6921	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 30718	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ)
6	3, 5	ax-mp 5	. 2 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
7		imsmetlem.2	. . . . . 6 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
8		imsmetlem.4	. . . . . 6 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
9		imsmetlem.6	. . . . . 6 ⊢ 𝑁 = (norm_CV‘𝑈)
10	1, 7, 8, 9, 4	imsdval2 30716	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
11	3, 10	mp3an1 1447	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
12	11	eqeq1d 2737	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0))
13		neg1cn 12378	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 30655	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1450	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (-1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 30649	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
17	3, 16	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
18	15, 17	sylan2 593	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
19		imsmetlem.5	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
20	1, 19, 9	nvz 30698	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
21	3, 18, 20	sylancr 587	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
22	1, 19	nvzcl 30663	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 30653	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
25	3, 24	mpan 690	. . . . . 6 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
26	23, 25	mp3an2 1448	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
27	18, 26	sylancom 588	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
28		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	15	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
30		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
31	1, 7	nvass 30651	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
32	3, 31	mpan 690	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
33	28, 29, 30, 32	syl3anc 1370	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
34	1, 7, 8, 19	nvlinv 30681	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
35	3, 34	mpan 690	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
36	35	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
37	36	oveq2d 7447	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 30664	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
39	3, 38	mpan 690	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐺𝑍) = 𝑥)
40	39	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
41	33, 37, 40	3eqtrd 2779	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = 𝑥)
42	1, 7, 19	nv0lid 30665	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
43	3, 42	mpan 690	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑍𝐺𝑦) = 𝑦)
44	43	adantl 481	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
45	41, 44	eqeq12d 2751	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ 𝑥 = 𝑦))
46	27, 45	bitr3d 281	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦)) = 𝑍 ↔ 𝑥 = 𝑦))
47	12, 21, 46	3bitrd 305	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
48		simpr 484	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 30655	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
50	3, 13, 49	mp3an12 1450	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → (-1𝑆𝑧) ∈ 𝑋)
51	50	adantr 480	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
52	1, 7	nvgcl 30649	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
53	3, 52	mp3an1 1447	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
54	48, 51, 53	syl2anc 584	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
55	54	3adant3 1131	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
56	1, 7	nvgcl 30649	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
57	3, 56	mp3an1 1447	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
58	15, 57	sylan2 593	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
59	58	3adant2 1130	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
60	1, 7, 9	nvtri 30699	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
61	3, 60	mp3an1 1447	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
62	55, 59, 61	syl2anc 584	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
63	11	3adant1 1129	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
64		simp1 1135	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑧 ∈ 𝑋)
65	15	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
66	1, 7	nvass 30651	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
67	3, 66	mpan 690	. . . . . . . 8 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
68	55, 64, 65, 67	syl3anc 1370	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
69		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑧 ∈ 𝑋)
70	1, 7	nvass 30651	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
71	3, 70	mpan 690	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
72	48, 51, 69, 71	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
73	1, 7, 8, 19	nvlinv 30681	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
74	3, 73	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
75	74	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
76	75	oveq2d 7447	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
77	39	adantl 481	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
78	72, 76, 77	3eqtrd 2779	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
79	78	3adant3 1131	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
80	79	oveq1d 7446	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
81	68, 80	eqtr3d 2777	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))) = (𝑥𝐺(-1𝑆𝑦)))
82	81	fveq2d 6911	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
83	63, 82	eqtr4d 2778	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))))
84	1, 7, 8, 9, 4	imsdval2 30716	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
85	3, 84	mp3an1 1447	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
86	1, 7, 8, 9	nvdif 30695	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
87	3, 86	mp3an1 1447	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
88	85, 87	eqtrd 2775	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
89	88	3adant3 1131	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
90	1, 7, 8, 9, 4	imsdval2 30716	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
91	3, 90	mp3an1 1447	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
92	91	3adant2 1130	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
93	89, 92	oveq12d 7449	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5180	. . 3 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
95	94	3coml 1126	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
96	2, 6, 47, 95	ismeti 24351	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ≤ cle 11294 -cneg 11491 Metcmet 21368 invcgn 30520 NrmCVeccnv 30613 +_𝑣 cpv 30614 BaseSetcba 30615 ·_𝑠OLD cns 30616 0_veccn0v 30617 norm_CVcnmcv 30619 IndMetcims 30620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-met 21376 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630

This theorem is referenced by: imsmet 30720

W3C validator