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Theorem imsmetlem 31292

Description: Lemma for imsmet 31293. (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 6899	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 31291	. . 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 31289	. . . . 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 2763	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ (𝑁‘(𝑥𝐺( -1𝑆𝑦))) = 0))
13		neg1cn 12305	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 31228	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → ( -1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1480	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → ( -1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 31222	. . . . . 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 605	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺( -1𝑆𝑦)) ∈ 𝑋)
19		imsmetlem.5	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑈)
20	1, 19, 9	nvz 31271	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝐺( -1𝑆𝑦)) ∈ 𝑋) → ((𝑁‘(𝑥𝐺( -1𝑆𝑦))) = 0 ↔ (𝑥𝐺( -1𝑆𝑦)) = 𝑍))
21	3, 18, 20	sylancr 599	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝐺( -1𝑆𝑦))) = 0 ↔ (𝑥𝐺( -1𝑆𝑦)) = 𝑍))
22	1, 19	nvzcl 31236	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 31226	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺( -1𝑆𝑦)) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐺( -1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺( -1𝑆𝑦)) = 𝑍))
25	3, 24	mpan 703	. . . . . 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 600	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺( -1𝑆𝑦))𝐺𝑦) = (𝑍𝐺𝑦) ↔ (𝑥𝐺( -1𝑆𝑦)) = 𝑍))
28		simpl 488	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	15	adantl 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ( -1𝑆𝑦) ∈ 𝑋)
30		simpr 490	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
31	1, 7	nvass 31224	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ ( -1𝑆𝑦) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐺( -1𝑆𝑦))𝐺𝑦) = (𝑥𝐺(( -1𝑆𝑦)𝐺𝑦)))
32	3, 31	mpan 703	. . . . . . 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 31254	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (( -1𝑆𝑦)𝐺𝑦) = 𝑍)
35	3, 34	mpan 703	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (( -1𝑆𝑦)𝐺𝑦) = 𝑍)
36	35	adantl 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (( -1𝑆𝑦)𝐺𝑦) = 𝑍)
37	36	oveq2d 7436	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(( -1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 31237	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
39	3, 38	mpan 703	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (𝑥𝐺𝑍) = 𝑥)
40	39	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
41	33, 37, 40	3eqtrd 2800	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺( -1𝑆𝑦))𝐺𝑦) = 𝑥)
42	1, 7, 19	nv0lid 31238	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
43	3, 42	mpan 703	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑍𝐺𝑦) = 𝑦)
44	43	adantl 487	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑍𝐺𝑦) = 𝑦)
45	41, 44	eqeq12d 2777	. . . 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 490	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 31228	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → ( -1𝑆𝑧) ∈ 𝑋)
50	3, 13, 49	mp3an12 1480	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑋 → ( -1𝑆𝑧) ∈ 𝑋)
51	50	adantr 486	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ( -1𝑆𝑧) ∈ 𝑋)
52	1, 7	nvgcl 31222	. . . . . . . 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 596	. . . . . 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 31222	. . . . . . . 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 605	. . . . . 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 31272	. . . . . 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 596	. . . 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 31224	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝑥𝐺( -1𝑆𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( -1𝑆𝑦) ∈ 𝑋)) → (((𝑥𝐺( -1𝑆𝑧))𝐺𝑧)𝐺( -1𝑆𝑦)) = ((𝑥𝐺( -1𝑆𝑧))𝐺(𝑧𝐺( -1𝑆𝑦))))
67	3, 66	mpan 703	. . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑧 ∈ 𝑋)
70	1, 7	nvass 31224	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ ( -1𝑆𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺( -1𝑆𝑧))𝐺𝑧) = (𝑥𝐺(( -1𝑆𝑧)𝐺𝑧)))
71	3, 70	mpan 703	. . . . . . . . . . 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 31254	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋) → (( -1𝑆𝑧)𝐺𝑧) = 𝑍)
74	3, 73	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 → (( -1𝑆𝑧)𝐺𝑧) = 𝑍)
75	74	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (( -1𝑆𝑧)𝐺𝑧) = 𝑍)
76	75	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺(( -1𝑆𝑧)𝐺𝑧)) = (𝑥𝐺𝑍))
77	39	adantl 487	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑍) = 𝑥)
78	72, 76, 77	3eqtrd 2800	. . . . . . . . 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 7435	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑥𝐺( -1𝑆𝑧))𝐺𝑧)𝐺( -1𝑆𝑦)) = (𝑥𝐺( -1𝑆𝑦)))
81	68, 80	eqtr3d 2798	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺( -1𝑆𝑧))𝐺(𝑧𝐺( -1𝑆𝑦))) = (𝑥𝐺( -1𝑆𝑦)))
82	81	fveq2d 6889	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥𝐺( -1𝑆𝑧))𝐺(𝑧𝐺( -1𝑆𝑦)))) = (𝑁‘(𝑥𝐺( -1𝑆𝑦))))
83	63, 82	eqtr4d 2799	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (𝑁‘((𝑥𝐺( -1𝑆𝑧))𝐺(𝑧𝐺( -1𝑆𝑦)))))
84	1, 7, 8, 9, 4	imsdval2 31289	. . . . . . . 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 31268	. . . . . . . 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 2796	. . . . . 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 31289	. . . . . . 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 7438	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺( -1𝑆𝑧))) + (𝑁‘(𝑧𝐺( -1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5137	. . 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 24644	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 × cxp 5649 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 ≤ cle 11344 -cneg 11542 Metcmet 21664 invcgn 31093 NrmCVeccnv 31186 +_𝑣 cpv 31187 BaseSetcba 31188 ·_𝑠OLD cns 31189 0_veccn0v 31190 norm_CVcnmcv 31192 IndMetcims 31193

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9434 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-seq 14145 df-exp 14205 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-met 21672 df-grpo 31095 df-gid 31096 df-ginv 31097 df-gdiv 31098 df-ablo 31147 df-vc 31161 df-nv 31194 df-va 31197 df-ba 31198 df-sm 31199 df-0v 31200 df-vs 31201 df-nmcv 31202 df-ims 31203

This theorem is used by: imsmet 31293

W3C validator