Theorem xmetres2 22971

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 6702	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2	1	adantr 483	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ dom ∞Met)
3		simpr 487	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4	2, 3	ssexd 5228	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
5		xmetf 22939	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	adantr 483	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7		xpss12 5570	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
8	3, 7	sylancom 590	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
9	6, 8	fssresd 6545	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
10		ovres 7314	. . . . 5 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
11	10	adantl 484	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
12	11	eqeq1d 2823	. . 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 3968	. . . 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 3968	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
19		xmeteq0 22948	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
20	13, 16, 18, 19	syl3anc 1367	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
21	12, 20	bitrd 281	. 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 1192	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑅)
25	23, 24	sseldd 3968	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑧 ∈ 𝑋)
26	16	3adantr3 1167	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑋)
27	18	3adantr3 1167	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑋)
28		xmettri2 22950	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
29	22, 25, 26, 27, 28	syl13anc 1368	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
30	11	3adantr3 1167	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑥(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑥𝐷𝑦))
31		simpr1 1190	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
32	24, 31	ovresd 7315	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑥) = (𝑧𝐷𝑥))
33		simpr2 1191	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
34	24, 33	ovresd 7315	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → (𝑧(𝐷 ↾ (𝑅 × 𝑅))𝑦) = (𝑧𝐷𝑦))
35	32, 34	oveq12d 7174	. . 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 5098	. 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 22936	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936   class class class wbr 5066   × cxp 5553  dom cdm 5555   ↾ cres 5557  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  0cc0 10537  ℝ^*cxr 10674   ≤ cle 10676   +_𝑒 cxad 12506  ∞Metcxmet 20530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-xr 10679  df-xmet 20538

This theorem is referenced by:  metres2  22973  xmetres  22974  xpsxmet  22990  xpsdsval  22991  xmetresbl  23047  tmsxms  23096  imasf1oxms  23099  metrest  23134  prdsxms  23140  tmsxpsval  23148  nrginvrcn  23301  divcn  23476  iitopon  23487  cncfmet  23516  cfilres  23899  dvlip2  24592  ftc1lem6  24638  ulmdvlem1  24988  ulmdvlem3  24990  abelth  25029  cxpcn3  25329  rlimcnp  25543  minvecolem4b  28655  minvecolem4  28657  ftc1cnnc  34981  blbnd  35080  ismtyres  35101  reheibor  35132