Theorem ressusp 22874

Description: The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Hypotheses

Ref	Expression
ressusp.1	⊢ 𝐵 = (Base‘𝑊)
ressusp.2	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
ressusp	⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (𝑊 ↾s 𝐴) ∈ UnifSp)

Proof of Theorem ressusp

Step	Hyp	Ref	Expression
1		ressuss 22872	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
2	1	3ad2ant3 1131	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
3		simp1 1132	. . . . . . 7 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝑊 ∈ UnifSp)
4		ressusp.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
5		eqid 2821	. . . . . . . 8 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊)
6		ressusp.2	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
7	4, 5, 6	isusp 22870	. . . . . . 7 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
8	3, 7	sylib 220	. . . . . 6 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
9	8	simpld 497	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (UnifSt‘𝑊) ∈ (UnifOn‘𝐵))
10		simp2 1133	. . . . . . 7 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝑊 ∈ TopSp)
11	4, 6	istps 21542	. . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
12	10, 11	sylib 220	. . . . . 6 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝐵))
13		simp3 1134	. . . . . 6 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ 𝐽)
14		toponss 21535	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝐵)
15	12, 13, 14	syl2anc 586	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝐵)
16		trust 22838	. . . . 5 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴 ⊆ 𝐵) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
17	9, 15, 16	syl2anc 586	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
18	2, 17	eqeltrd 2913	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (UnifSt‘(𝑊 ↾s 𝐴)) ∈ (UnifOn‘𝐴))
19		eqid 2821	. . . . . 6 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
20	19, 4	ressbas2 16555	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
21	15, 20	syl 17	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
22	21	fveq2d 6674	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (UnifOn‘𝐴) = (UnifOn‘(Base‘(𝑊 ↾s 𝐴))))
23	18, 22	eleqtrd 2915	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (UnifSt‘(𝑊 ↾s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊 ↾s 𝐴))))
24	8	simprd 498	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐽 = (unifTop‘(UnifSt‘𝑊)))
25	13, 24	eleqtrd 2915	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ (unifTop‘(UnifSt‘𝑊)))
26		restutopopn 22847	. . . 4 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴 ∈ (unifTop‘(UnifSt‘𝑊))) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
27	9, 25, 26	syl2anc 586	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
28	24	oveq1d 7171	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (𝐽 ↾t 𝐴) = ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴))
29	2	fveq2d 6674	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (unifTop‘(UnifSt‘(𝑊 ↾s 𝐴))) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
30	27, 28, 29	3eqtr4d 2866	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (𝐽 ↾t 𝐴) = (unifTop‘(UnifSt‘(𝑊 ↾s 𝐴))))
31		eqid 2821	. . 3 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
32		eqid 2821	. . 3 ⊢ (UnifSt‘(𝑊 ↾s 𝐴)) = (UnifSt‘(𝑊 ↾s 𝐴))
33	19, 6	resstopn 21794	. . 3 ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘(𝑊 ↾s 𝐴))
34	31, 32, 33	isusp 22870	. 2 ⊢ ((𝑊 ↾s 𝐴) ∈ UnifSp ↔ ((UnifSt‘(𝑊 ↾s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊 ↾s 𝐴))) ∧ (𝐽 ↾t 𝐴) = (unifTop‘(UnifSt‘(𝑊 ↾s 𝐴)))))
35	23, 30, 34	sylanbrc 585	1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (𝑊 ↾s 𝐴) ∈ UnifSp)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936   × cxp 5553  ‘cfv 6355  (class class class)co 7156  Basecbs 16483   ↾_s cress 16484   ↾_t crest 16694  TopOpenctopn 16695  TopOnctopon 21518  TopSpctps 21540  UnifOncust 22808  unifTopcutop 22839  UnifStcuss 22862  UnifSpcusp 22863

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-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  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-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-tset 16584  df-unif 16588  df-rest 16696  df-topn 16697  df-top 21502  df-topon 21519  df-topsp 21541  df-ust 22809  df-utop 22840  df-uss 22865  df-usp 22866

This theorem is referenced by: (None)