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Theorem imsmetlem 30894

Description: Lemma for imsmet 30895. (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 6882	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 30893	. . 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 30891	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
11	3, 10	mp3an1 1470	. . . 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 12181	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 30830	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1473	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (-1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 30824	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
17	3, 16	mp3an1 1470	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
18	15, 17	sylan2 602	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋)
19		imsmetlem.5	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
20	1, 19, 9	nvz 30873	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
21	3, 18, 20	sylancr 596	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝐺(-1𝑆𝑦))) = 0 ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
22	1, 19	nvzcl 30838	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 30828	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
25	3, 24	mpan 700	. . . . . 6 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
26	23, 25	mp3an2 1471	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑦)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
27	18, 26	sylancom 597	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺(-1𝑆𝑦)) = 𝑍))
28		simpl 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	15	adantl 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
30		simpr 488	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
31	1, 7	nvass 30826	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
32	3, 31	mpan 700	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
33	28, 29, 30, 32	syl3anc 1391	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦))𝐺𝑦) = (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)))
34	1, 7, 8, 19	nvlinv 30856	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
35	3, 34	mpan 700	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
36	35	adantl 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((-1𝑆𝑦)𝐺𝑦) = 𝑍)
37	36	oveq2d 7413	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 30839	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
39	3, 38	mpan 700	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐺𝑍) = 𝑥)
40	39	adantr 484	. . . . . 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 30840	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
43	3, 42	mpan 700	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑍𝐺𝑦) = 𝑦)
44	43	adantl 485	. . . . 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 283	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑦)) = 𝑍 ↔ 𝑥 = 𝑦))
47	12, 21, 46	3bitrd 307	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
48		simpr 488	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 30830	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
50	3, 13, 49	mp3an12 1473	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → (-1𝑆𝑧) ∈ 𝑋)
51	50	adantr 484	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (-1𝑆𝑧) ∈ 𝑋)
52	1, 7	nvgcl 30824	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
53	3, 52	mp3an1 1470	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
54	48, 51, 53	syl2anc 593	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
55	54	3adant3 1146	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋)
56	1, 7	nvgcl 30824	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
57	3, 56	mp3an1 1470	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
58	15, 57	sylan2 602	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
59	58	3adant2 1145	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋)
60	1, 7, 9	nvtri 30874	. . . . . 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 1470	. . . . 5 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ (𝑧𝐺(-1𝑆𝑦)) ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
62	55, 59, 61	syl2anc 593	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦)))) ≤ ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
63	11	3adant1 1144	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
64		simp1 1150	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑧 ∈ 𝑋)
65	15	3ad2ant3 1149	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) ∈ 𝑋)
66	1, 7	nvass 30826	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
67	3, 66	mpan 700	. . . . . . . 8 ⊢ (((𝑥𝐺(-1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (-1𝑆𝑦) ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
68	55, 64, 65, 67	syl3anc 1391	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺(-1𝑆𝑧))𝐺𝑧)𝐺(-1𝑆𝑦)) = ((𝑥𝐺(-1𝑆𝑧))𝐺(𝑧𝐺(-1𝑆𝑦))))
69		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑧 ∈ 𝑋)
70	1, 7	nvass 30826	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
71	3, 70	mpan 700	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (-1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
72	48, 51, 69, 71	syl3anc 1391	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)))
73	1, 7, 8, 19	nvlinv 30856	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
74	3, 73	mpan 700	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
75	74	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((-1𝑆𝑧)𝐺𝑧) = 𝑍)
76	75	oveq2d 7413	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺((-1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
77	39	adantl 485	. . . . . . . . . 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 1146	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺(-1𝑆𝑧))𝐺𝑧) = 𝑥)
80	79	oveq1d 7412	. . . . . . 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 6872	. . . . 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 30891	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
85	3, 84	mp3an1 1470	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑧𝐺(-1𝑆𝑥))))
86	1, 7, 8, 9	nvdif 30870	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑧𝐺(-1𝑆𝑥))) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
87	3, 86	mp3an1 1470	. . . . . . 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 1146	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑥) = (𝑁‘(𝑥𝐺(-1𝑆𝑧))))
90	1, 7, 8, 9, 4	imsdval2 30891	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
91	3, 90	mp3an1 1470	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
92	91	3adant2 1145	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (𝑁‘(𝑧𝐺(-1𝑆𝑦))))
93	89, 92	oveq12d 7415	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺(-1𝑆𝑧))) + (𝑁‘(𝑧𝐺(-1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5133	. . 3 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
95	94	3coml 1141	. 2 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
96	2, 6, 47, 95	ismeti 24386	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 class class class wbr 5101 × cxp 5646 ⟶wf 6518 ‘cfv 6522 (class class class)co 7397 ℂcc 11072 ℝcr 11073 0cc0 11074 1c1 11075 + caddc 11077 ≤ cle 11218 -cneg 11416 Metcmet 21411 invcgn 30695 NrmCVeccnv 30788 +_𝑣 cpv 30789 BaseSetcba 30790 ·_𝑠OLD cns 30791 0_veccn0v 30792 norm_CVcnmcv 30794 IndMetcims 30795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-er 8679 df-map 8811 df-en 8929 df-dom 8930 df-sdom 8931 df-sup 9389 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-n0 12483 df-z 12570 df-uz 12841 df-rp 12995 df-seq 14016 df-exp 14076 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-met 21419 df-grpo 30697 df-gid 30698 df-ginv 30699 df-gdiv 30700 df-ablo 30749 df-vc 30763 df-nv 30796 df-va 30799 df-ba 30800 df-sm 30801 df-0v 30802 df-vs 30803 df-nmcv 30804 df-ims 30805

This theorem is referenced by: imsmet 30895

W3C validator