Theorem psmetres2 13500

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 13492	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2	1	adantr 276	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3		simpr 110	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4		xpss12 4730	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
5	3, 3, 4	syl2anc 411	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
6	2, 5	fssresd 5388	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7		simpr 110	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑅)
8	7, 7	ovresd 6009	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9		simpll 527	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
10	3	sselda 3155	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑋)
11		psmet0 13494	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
12	9, 10, 11	syl2anc 411	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎𝐷𝑎) = 0)
13	8, 12	eqtrd 2210	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
14	9	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
15	3	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
16	15	sselda 3155	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑋)
17	10	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑋)
18	3	adantr 276	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
19	18	sselda 3155	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑋)
20	19	adantr 276	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑋)
21		psmettri2 13495	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
22	14, 16, 17, 20, 21	syl13anc 1240	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
23	7	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
24		simpr 110	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
25	23, 24	ovresd 6009	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
26	25	adantr 276	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27		simpr 110	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑅)
28	7	ad2antrr 488	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑅)
29	27, 28	ovresd 6009	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
30	24	adantr 276	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑅)
31	27, 30	ovresd 6009	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
32	29, 31	oveq12d 5887	. . . . . . 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 4032	. . . . . 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 2550	. . . . 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 2550	. . . 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 306	. . 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 2550	. 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 13154	. . . . . 6 ⊢ PsMet = (𝑎 ∈ V ↦ {𝑏 ∈ (ℝ* ↑𝑚 (𝑎 × 𝑎)) ∣ ∀𝑐 ∈ 𝑎 ((𝑐𝑏𝑐) = 0 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑎 (𝑐𝑏𝑑) ≤ ((𝑒𝑏𝑐) +𝑒 (𝑒𝑏𝑑)))})
39	38	mptrcl 5594	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
40	39	adantr 276	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ V)
41	40, 3	ssexd 4140	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
42		ispsmet 13490	. . 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 944	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {crab 2459  Vcvv 2737   ⊆ wss 3129   class class class wbr 4000   × cxp 4621   ↾ cres 4625  ⟶wf 5208  ‘cfv 5212  (class class class)co 5869   ↑_𝑚 cmap 6642  0cc0 7802  ℝ^*cxr 7981   ≤ cle 7983   +_𝑒 cxad 9757  PsMetcpsmet 13146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-map 6644  df-pnf 7984  df-mnf 7985  df-xr 7986  df-psmet 13154

This theorem is referenced by: (None)