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Theorem imsmetlem 31307

Description: Lemma for imsmet 31308. (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 6910	. 2 ⊢ 𝑋 ∈ V
3		imsmetlem.9	. . 3 ⊢ 𝑈 ∈ NrmCVec
4		imsmetlem.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
5	1, 4	imsdf 31306	. . 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 31304	. . . . 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 12318	. . . . . 6 ⊢ -1 ∈ ℂ
14	1, 8	nvscl 31243	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → ( -1𝑆𝑦) ∈ 𝑋)
15	3, 13, 14	mp3an12 1480	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → ( -1𝑆𝑦) ∈ 𝑋)
16	1, 7	nvgcl 31237	. . . . . 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 31286	. . . 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 31251	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
23	3, 22	ax-mp 5	. . . . . 6 ⊢ 𝑍 ∈ 𝑋
24	1, 7	nvrcan 31241	. . . . . . 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 31239	. . . . . . . 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 31269	. . . . . . . . 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 7446	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(( -1𝑆𝑦)𝐺𝑦)) = (𝑥𝐺𝑍))
38	1, 7, 19	nv0rid 31252	. . . . . . . 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 2802	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐺( -1𝑆𝑦))𝐺𝑦) = 𝑥)
42	1, 7, 19	nv0lid 31253	. . . . . . 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 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 490	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
49	1, 8	nvscl 31243	. . . . . . . . 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 31237	. . . . . . . 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 31237	. . . . . . . 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 31287	. . . . . 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 31239	. . . . . . . . 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 31239	. . . . . . . . . . . 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 31269	. . . . . . . . . . . . 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 7446	. . . . . . . . . 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 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 7445	. . . . . . 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 6900	. . . . 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 31304	. . . . . . . 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 31283	. . . . . . . 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 31304	. . . . . . 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 7448	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑁‘(𝑥𝐺( -1𝑆𝑧))) + (𝑁‘(𝑧𝐺( -1𝑆𝑦)))))
94	62, 83, 93	3brtr4d 5140	. . 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 24659	1 ⊢ 𝐷 ∈ (Met‘𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 class class class wbr 5106 × cxp 5653 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ℂcc 11211 ℝcr 11212 0cc0 11213 1c1 11214 + caddc 11216 ≤ cle 11357 -cneg 11555 Metcmet 21679 invcgn 31108 NrmCVeccnv 31201 +_𝑣 cpv 31202 BaseSetcba 31203 ·_𝑠OLD cns 31204 0_veccn0v 31205 norm_CVcnmcv 31207 IndMetcims 31208

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-sup 9447 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-n0 12620 df-z 12707 df-uz 12979 df-rp 13134 df-seq 14158 df-exp 14218 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-met 21687 df-grpo 31110 df-gid 31111 df-ginv 31112 df-gdiv 31113 df-ablo 31162 df-vc 31176 df-nv 31209 df-va 31212 df-ba 31213 df-sm 31214 df-0v 31215 df-vs 31216 df-nmcv 31217 df-ims 31218

This theorem is used by: imsmet 31308

W3C validator