Theorem xmetres2 24256

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 6898	. . . 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 5282	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
5		xmetf 24224	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	adantr 480	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7		xpss12 5656	. . . 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 6730	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
10		ovres 7558	. . . . 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 2732	. . 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 3950	. . . 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 3950	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
19		xmeteq0 24233	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
20	13, 16, 18, 19	syl3anc 1373	. . 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 1197	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑅)
25	23, 24	sseldd 3950	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑋)
26	16	3adantr3 1172	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
27	18	3adantr3 1172	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
28		xmettri2 24235	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
29	22, 25, 26, 27, 28	syl13anc 1374	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
30	11	3adantr3 1172	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
31		simpr1 1195	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
32	24, 31	ovresd 7559	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) = (𝑧𝐷𝑥))
33		simpr2 1196	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
34	24, 33	ovresd 7559	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑧𝐷𝑦))
35	32, 34	oveq12d 7408	. . 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 5142	. 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 24221	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917   class class class wbr 5110   × cxp 5639  dom cdm 5641   ↾ cres 5643  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  0cc0 11075  ℝ^*cxr 11214   ≤ cle 11216   +_𝑒 cxad 13077  ∞Metcxmet 21256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-xr 11219  df-xmet 21264

This theorem is referenced by:  metres2  24258  xmetres  24259  xpsxmet  24275  xpsdsval  24276  xmetresbl  24332  tmsxms  24381  imasf1oxms  24384  metrest  24419  prdsxms  24425  tmsxpsval  24433  nrginvrcn  24587  divcnOLD  24764  divcn  24766  iitopon  24779  cncfmet  24809  cfilres  25203  dvlip2  25907  ftc1lem6  25955  ulmdvlem1  26316  ulmdvlem3  26318  abelth  26358  cxpcn3  26665  rlimcnp  26882  minvecolem4b  30814  minvecolem4  30816  ftc1cnnc  37693  blbnd  37788  ismtyres  37809  reheibor  37840