Theorem psmetres2 24301

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psmetf 24293	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2	1	adantr 482	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3		simpr 486	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4		xpss12 5636	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
5	3, 3, 4	syl2anc 591	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
6	2, 5	fssresd 6698	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7		simpr 486	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑅)
8	7, 7	ovresd 7527	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9		simpll 773	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
10	3	sselda 3917	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑋)
11		psmet0 24295	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
12	9, 10, 11	syl2anc 591	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎𝐷𝑎) = 0)
13	8, 12	eqtrd 2776	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
14	9	ad2antrr 733	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
15	3	ad2antrr 733	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
16	15	sselda 3917	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑋)
17	10	ad2antrr 733	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑋)
18	3	adantr 482	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
19	18	sselda 3917	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑋)
20	19	adantr 482	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑋)
21		psmettri2 24296	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
22	14, 16, 17, 20, 21	syl13anc 1381	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
23	7	adantr 482	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
24		simpr 486	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
25	23, 24	ovresd 7527	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
26	25	adantr 482	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27		simpr 486	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑅)
28	7	ad2antrr 733	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑅)
29	27, 28	ovresd 7527	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
30	24	adantr 482	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑅)
31	27, 30	ovresd 7527	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
32	29, 31	oveq12d 7378	. . . . . . 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 5107	. . . . . 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 3133	. . . . 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 3133	. . . 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 517	. . 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 3133	. 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		elfvex 6866	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39	38	adantr 482	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ V)
40	39, 3	ssexd 5255	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
41		ispsmet 24291	. . 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))))))
42	40, 41	syl 17	. 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))))))
43	6, 37, 42	mpbir2and 720	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3885   class class class wbr 5075   × cxp 5619   ↾ cres 5623  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  0cc0 11033  ℝ^*cxr 11173   ≤ cle 11175   +_𝑒 cxad 13056  PsMetcpsmet 21335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-xr 11178  df-psmet 21343

This theorem is referenced by:  restmetu  24557