Theorem xmetres2 24311

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 6933	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2	1	adantr 479	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ dom ∞Met)
3		simpr 483	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4	2, 3	ssexd 5325	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
5		xmetf 24279	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	adantr 479	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7		xpss12 5693	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
8	3, 7	sylancom 586	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
9	6, 8	fssresd 6764	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
10		ovres 7587	. . . . 5 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
11	10	adantl 480	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
12	11	eqeq1d 2727	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
13		simpll 765	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
14		simplr 767	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑅 ⊆ 𝑋)
15		simprl 769	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
16	14, 15	sseldd 3977	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
17		simprr 771	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
18	14, 17	sseldd 3977	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
19		xmeteq0 24288	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
20	13, 16, 18, 19	syl3anc 1368	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
21	12, 20	bitrd 278	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = 0 ↔ 𝑥 = 𝑦))
22		simpll 765	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
23		simplr 767	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑅 ⊆ 𝑋)
24		simpr3 1193	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑅)
25	23, 24	sseldd 3977	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑋)
26	16	3adantr3 1168	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
27	18	3adantr3 1168	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
28		xmettri2 24290	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
29	22, 25, 26, 27, 28	syl13anc 1369	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
30	11	3adantr3 1168	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
31		simpr1 1191	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
32	24, 31	ovresd 7588	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) = (𝑧𝐷𝑥))
33		simpr2 1192	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
34	24, 33	ovresd 7588	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑧𝐷𝑦))
35	32, 34	oveq12d 7437	. . 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 5181	. 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 24276	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ⊆ wss 3944   class class class wbr 5149   × cxp 5676  dom cdm 5678   ↾ cres 5680  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  0cc0 11140  ℝ^*cxr 11279   ≤ cle 11281   +_𝑒 cxad 13125  ∞Metcxmet 21281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-xr 11284  df-xmet 21289

This theorem is referenced by:  metres2  24313  xmetres  24314  xpsxmet  24330  xpsdsval  24331  xmetresbl  24387  tmsxms  24439  imasf1oxms  24442  metrest  24477  prdsxms  24483  tmsxpsval  24491  nrginvrcn  24653  divcnOLD  24828  divcn  24830  iitopon  24843  cncfmet  24873  cfilres  25268  dvlip2  25972  ftc1lem6  26020  ulmdvlem1  26381  ulmdvlem3  26383  abelth  26423  cxpcn3  26728  rlimcnp  26942  minvecolem4b  30760  minvecolem4  30762  ftc1cnnc  37296  blbnd  37391  ismtyres  37412  reheibor  37443