imsmetlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > imsmetlem		Structured version Visualization version GIF version

Theorem imsmetlem 29046

Description: Lemma for imsmet 29047. (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 6783	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 29045	. . 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 29043	. . . . 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 2742	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0))
13		neg1cn 12085	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 28982	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1450	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (-1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 28976	. . . . . 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 29025	. . . 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 28990	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 28980	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
25	3, 24	mpan 687	. . . . . 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 483	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	15	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
30		simpr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
31	1, 7	nvass 28978	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
32	3, 31	mpan 687	. . . . . . 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 29008	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
35	3, 34	mpan 687	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
36	35	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
37	36	oveq2d 7285	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 28991	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
39	3, 38	mpan 687	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐺𝑍) = 𝑥)
40	39	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
41	33, 37, 40	3eqtrd 2784	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = 𝑥)
42	1, 7, 19	nv0lid 28992	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
43	3, 42	mpan 687	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑍𝐺𝑦) = 𝑦)
44	43	adantl 482	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
45	41, 44	eqeq12d 2756	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ 𝑥 = 𝑦))
46	27, 45	bitr3d 280	. . 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 485	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 28982	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
50	3, 13, 49	mp3an12 1450	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → (-1𝑆𝑧) ∈ 𝑋)
51	50	adantr 481	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
52	1, 7	nvgcl 28976	. . . . . . . 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 28976	. . . . . . . 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 29026	. . . . . 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 28978	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
67	3, 66	mpan 687	. . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑧 ∈ 𝑋)
70	1, 7	nvass 28978	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
71	3, 70	mpan 687	. . . . . . . . . . 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 29008	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
74	3, 73	mpan 687	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
75	74	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
76	75	oveq2d 7285	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
77	39	adantl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
78	72, 76, 77	3eqtrd 2784	. . . . . . . . 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 7284	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
81	68, 80	eqtr3d 2782	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))) = (𝑥𝐺(-1𝑆𝑦)))
82	81	fveq2d 6773	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
83	63, 82	eqtr4d 2783	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))))
84	1, 7, 8, 9, 4	imsdval2 29043	. . . . . . . 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 29022	. . . . . . . 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 2780	. . . . . 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 29043	. . . . . . 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 7287	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5111	. . 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 23474	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 × cxp 5587 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 ℝcr 10869 0cc0 10870 1c1 10871 + caddc 10873 ≤ cle 11009 -cneg 11204 Metcmet 20579 invcgn 28847 NrmCVeccnv 28940 +_𝑣 cpv 28941 BaseSetcba 28942 ·_𝑠OLD cns 28943 0_veccn0v 28944 norm_CVcnmcv 28946 IndMetcims 28947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 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 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-sup 9177 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12580 df-rp 12728 df-seq 13718 df-exp 13779 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-met 20587 df-grpo 28849 df-gid 28850 df-ginv 28851 df-gdiv 28852 df-ablo 28901 df-vc 28915 df-nv 28948 df-va 28951 df-ba 28952 df-sm 28953 df-0v 28954 df-vs 28955 df-nmcv 28956 df-ims 28957

This theorem is referenced by: imsmet 29047

W3C validator