psmetres2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > psmetres2		GIF version

Theorem psmetres2 13615

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 13607	. . . 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 4731	. . . 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 5389	. 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 6010	. . . . 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 13609	. . . . . 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 13610	. . . . . . . 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 6010	. . . . . . . 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 6010	. . . . . . . 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 6010	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
32	29, 31	oveq12d 5888	. . . . . . 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 4033	. . . . . 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 13252	. . . . . 6 ⊢ PsMet = (𝑎 ∈ V ↦ {𝑏 ∈ (ℝ^* ↑_𝑚 (𝑎 × 𝑎)) ∣ ∀𝑐 ∈ 𝑎 ((𝑐𝑏𝑐) = 0 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑎 (𝑐𝑏𝑑) ≤ ((𝑒𝑏𝑐) +_𝑒 (𝑒𝑏𝑑)))})
39	38	mptrcl 5595	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
40	39	adantr 276	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ V)
41	40, 3	ssexd 4141	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
42		ispsmet 13605	. . 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))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 Vcvv 2737 ⊆ wss 3129 class class class wbr 4001 × cxp 4622 ↾ cres 4626 ⟶wf 5209 ‘cfv 5213 (class class class)co 5870 ↑_𝑚 cmap 6643 0cc0 7806 ℝ^*cxr 7985 ≤ cle 7987 +_𝑒 cxad 9764 PsMetcpsmet 13244

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 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-cnex 7897 ax-resscn 7898

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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-br 4002 df-opab 4063 df-mpt 4064 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-fv 5221 df-ov 5873 df-oprab 5874 df-mpo 5875 df-map 6645 df-pnf 7988 df-mnf 7989 df-xr 7990 df-psmet 13252

This theorem is referenced by: (None)

W3C validator