Theorem xmetres2 24372

Description: Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmetres2	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Proof of Theorem xmetres2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6942	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2	1	adantr 480	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ dom ∞Met)
3		simpr 484	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4	2, 3	ssexd 5323	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
5		xmetf 24340	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	adantr 480	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7		xpss12 5699	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
8	3, 7	sylancom 588	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
9	6, 8	fssresd 6774	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
10		ovres 7600	. . . . 5 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
11	10	adantl 481	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
12	11	eqeq1d 2738	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
13		simpll 766	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
14		simplr 768	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑅 ⊆ 𝑋)
15		simprl 770	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
16	14, 15	sseldd 3983	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
17		simprr 772	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
18	14, 17	sseldd 3983	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
19		xmeteq0 24349	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
20	13, 16, 18, 19	syl3anc 1372	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
21	12, 20	bitrd 279	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = 0 ↔ 𝑥 = 𝑦))
22		simpll 766	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
23		simplr 768	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑅 ⊆ 𝑋)
24		simpr3 1196	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑅)
25	23, 24	sseldd 3983	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑋)
26	16	3adantr3 1171	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
27	18	3adantr3 1171	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
28		xmettri2 24351	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
29	22, 25, 26, 27, 28	syl13anc 1373	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
30	11	3adantr3 1171	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
31		simpr1 1194	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
32	24, 31	ovresd 7601	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) = (𝑧𝐷𝑥))
33		simpr2 1195	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
34	24, 33	ovresd 7601	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑧𝐷𝑦))
35	32, 34	oveq12d 7450	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ((𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) +𝑒 (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
36	29, 30, 35	3brtr4d 5174	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) ≤ ((𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) +𝑒 (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦)))
37	4, 9, 21, 36	isxmetd 24337	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950   class class class wbr 5142   × cxp 5682  dom cdm 5684   ↾ cres 5686  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  0cc0 11156  ℝ^*cxr 11295   ≤ cle 11297   +_𝑒 cxad 13153  ∞Metcxmet 21350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-xr 11300  df-xmet 21358

This theorem is referenced by:  metres2  24374  xmetres  24375  xpsxmet  24391  xpsdsval  24392  xmetresbl  24448  tmsxms  24500  imasf1oxms  24503  metrest  24538  prdsxms  24544  tmsxpsval  24552  nrginvrcn  24714  divcnOLD  24891  divcn  24893  iitopon  24906  cncfmet  24936  cfilres  25331  dvlip2  26035  ftc1lem6  26083  ulmdvlem1  26444  ulmdvlem3  26446  abelth  26486  cxpcn3  26792  rlimcnp  27009  minvecolem4b  30898  minvecolem4  30900  ftc1cnnc  37700  blbnd  37795  ismtyres  37816  reheibor  37847