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Theorem imsmetlem 31042

Description: Lemma for imsmet 31043. (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 6895	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 31041	. . 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 31039	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
11	3, 10	mp3an1 1477	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
12	11	eqeq1d 2765	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0))
13		neg1cn 12198	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 30978	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1480	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (-1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 30972	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
17	3, 16	mp3an1 1477	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
18	15, 17	sylan2 604	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
19		imsmetlem.5	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
20	1, 19, 9	nvz 31021	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
21	3, 18, 20	sylancr 598	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
22	1, 19	nvzcl 30986	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 30976	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
25	3, 24	mpan 702	. . . . . 6 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
26	23, 25	mp3an2 1478	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
27	18, 26	sylancom 599	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
28		simpl 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	15	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
30		simpr 489	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
31	1, 7	nvass 30974	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
32	3, 31	mpan 702	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
33	28, 29, 30, 32	syl3anc 1398	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
34	1, 7, 8, 19	nvlinv 31004	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
35	3, 34	mpan 702	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
36	35	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
37	36	oveq2d 7426	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 30987	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
39	3, 38	mpan 702	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐺𝑍) = 𝑥)
40	39	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
41	33, 37, 40	3eqtrd 2802	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = 𝑥)
42	1, 7, 19	nv0lid 30988	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
43	3, 42	mpan 702	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑍𝐺𝑦) = 𝑦)
44	43	adantl 486	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
45	41, 44	eqeq12d 2779	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ 𝑥 = 𝑦))
46	27, 45	bitr3d 284	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦)) = 𝑍 ↔ 𝑥 = 𝑦))
47	12, 21, 46	3bitrd 308	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
48		simpr 489	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 30978	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
50	3, 13, 49	mp3an12 1480	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → (-1𝑆𝑧) ∈ 𝑋)
51	50	adantr 485	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
52	1, 7	nvgcl 30972	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
53	3, 52	mp3an1 1477	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
54	48, 51, 53	syl2anc 595	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
55	54	3adant3 1150	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
56	1, 7	nvgcl 30972	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
57	3, 56	mp3an1 1477	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
58	15, 57	sylan2 604	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
59	58	3adant2 1149	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
60	1, 7, 9	nvtri 31022	. . . . . 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 1477	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
62	55, 59, 61	syl2anc 595	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
63	11	3adant1 1148	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
64		simp1 1154	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑧 ∈ 𝑋)
65	15	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
66	1, 7	nvass 30974	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
67	3, 66	mpan 702	. . . . . . . 8 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
68	55, 64, 65, 67	syl3anc 1398	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
69		simpl 487	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑧 ∈ 𝑋)
70	1, 7	nvass 30974	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
71	3, 70	mpan 702	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
72	48, 51, 69, 71	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
73	1, 7, 8, 19	nvlinv 31004	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
74	3, 73	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
75	74	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
76	75	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
77	39	adantl 486	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
78	72, 76, 77	3eqtrd 2802	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
79	78	3adant3 1150	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
80	79	oveq1d 7425	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
81	68, 80	eqtr3d 2800	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))) = (𝑥𝐺(-1𝑆𝑦)))
82	81	fveq2d 6885	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
83	63, 82	eqtr4d 2801	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))))
84	1, 7, 8, 9, 4	imsdval2 31039	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
85	3, 84	mp3an1 1477	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
86	1, 7, 8, 9	nvdif 31018	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
87	3, 86	mp3an1 1477	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
88	85, 87	eqtrd 2798	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
89	88	3adant3 1150	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
90	1, 7, 8, 9, 4	imsdval2 31039	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
91	3, 90	mp3an1 1477	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
92	91	3adant2 1149	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
93	89, 92	oveq12d 7428	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5143	. . 3 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
95	94	3coml 1145	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
96	2, 6, 47, 95	ismeti 24482	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 class class class wbr 5109 × cxp 5659 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 ≤ cle 11239 -cneg 11437 Metcmet 21508 invcgn 30843 NrmCVeccnv 30936 +_𝑣 cpv 30937 BaseSetcba 30938 ·_𝑠OLD cns 30939 0_veccn0v 30940 norm_CVcnmcv 30942 IndMetcims 30943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-met 21516 df-grpo 30845 df-gid 30846 df-ginv 30847 df-gdiv 30848 df-ablo 30897 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-vs 30951 df-nmcv 30952 df-ims 30953

This theorem is referenced by: imsmet 31043

W3C validator