Theorem psmetres2 12744

Description: Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
psmetres2	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Proof of Theorem psmetres2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psmetf 12736	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2	1	adantr 274	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3		simpr 109	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4		xpss12 4693	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
5	3, 3, 4	syl2anc 409	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
6	2, 5	fssresd 5346	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7		simpr 109	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑅)
8	7, 7	ovresd 5961	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9		simpll 519	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
10	3	sselda 3128	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑋)
11		psmet0 12738	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
12	9, 10, 11	syl2anc 409	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎𝐷𝑎) = 0)
13	8, 12	eqtrd 2190	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
14	9	ad2antrr 480	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
15	3	ad2antrr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
16	15	sselda 3128	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑋)
17	10	ad2antrr 480	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑋)
18	3	adantr 274	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
19	18	sselda 3128	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑋)
20	19	adantr 274	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑋)
21		psmettri2 12739	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
22	14, 16, 17, 20, 21	syl13anc 1222	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
23	7	adantr 274	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
24		simpr 109	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
25	23, 24	ovresd 5961	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
26	25	adantr 274	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27		simpr 109	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑅)
28	7	ad2antrr 480	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑅)
29	27, 28	ovresd 5961	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
30	24	adantr 274	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑅)
31	27, 30	ovresd 5961	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
32	29, 31	oveq12d 5842	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
33	22, 26, 32	3brtr4d 3996	. . . . . 6 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
34	33	ralrimiva 2530	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
35	34	ralrimiva 2530	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
36	13, 35	jca 304	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
37	36	ralrimiva 2530	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
38		df-psmet 12398	. . . . . 6 ⊢ PsMet = (𝑎 ∈ V ↦ {𝑏 ∈ (ℝ* ↑𝑚 (𝑎 × 𝑎)) ∣ ∀𝑐 ∈ 𝑎 ((𝑐𝑏𝑐) = 0 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑎 (𝑐𝑏𝑑) ≤ ((𝑒𝑏𝑐) +𝑒 (𝑒𝑏𝑑)))})
39	38	mptrcl 5550	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
40	39	adantr 274	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ V)
41	40, 3	ssexd 4104	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
42		ispsmet 12734	. . 3 ⊢ (𝑅 ∈ V → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
43	41, 42	syl 14	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
44	6, 37, 43	mpbir2and 929	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∀wral 2435  {crab 2439  Vcvv 2712   ⊆ wss 3102   class class class wbr 3965   × cxp 4584   ↾ cres 4588  ⟶wf 5166  ‘cfv 5170  (class class class)co 5824   ↑_𝑚 cmap 6593  0cc0 7732  ℝ^*cxr 7911   ≤ cle 7913   +_𝑒 cxad 9677  PsMetcpsmet 12390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-map 6595  df-pnf 7914  df-mnf 7915  df-xr 7916  df-psmet 12398

This theorem is referenced by: (None)