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Theorem ressusp 21979
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 ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)

Proof of Theorem ressusp
StepHypRef Expression
1 ressuss 21977 . . . . 5 (𝐴𝐽 → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
213ad2ant3 1082 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
3 simp1 1059 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ UnifSp)
4 ressusp.1 . . . . . . . 8 𝐵 = (Base‘𝑊)
5 eqid 2621 . . . . . . . 8 (UnifSt‘𝑊) = (UnifSt‘𝑊)
6 ressusp.2 . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
74, 5, 6isusp 21975 . . . . . . 7 (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
83, 7sylib 208 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
98simpld 475 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘𝑊) ∈ (UnifOn‘𝐵))
10 simp2 1060 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ TopSp)
114, 6istps 20651 . . . . . . 7 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
1210, 11sylib 208 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 ∈ (TopOn‘𝐵))
13 simp3 1061 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐽)
14 toponss 20644 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐴𝐽) → 𝐴𝐵)
1512, 13, 14syl2anc 692 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐵)
16 trust 21943 . . . . 5 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴𝐵) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
179, 15, 16syl2anc 692 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
182, 17eqeltrd 2698 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘𝐴))
19 eqid 2621 . . . . . 6 (𝑊s 𝐴) = (𝑊s 𝐴)
2019, 4ressbas2 15852 . . . . 5 (𝐴𝐵𝐴 = (Base‘(𝑊s 𝐴)))
2115, 20syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 = (Base‘(𝑊s 𝐴)))
2221fveq2d 6152 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifOn‘𝐴) = (UnifOn‘(Base‘(𝑊s 𝐴))))
2318, 22eleqtrd 2700 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))))
248simprd 479 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 = (unifTop‘(UnifSt‘𝑊)))
2513, 24eleqtrd 2700 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 ∈ (unifTop‘(UnifSt‘𝑊)))
26 restutopopn 21952 . . . 4 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴 ∈ (unifTop‘(UnifSt‘𝑊))) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
279, 25, 26syl2anc 692 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
2824oveq1d 6619 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴))
292fveq2d 6152 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (unifTop‘(UnifSt‘(𝑊s 𝐴))) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
3027, 28, 293eqtr4d 2665 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴))))
31 eqid 2621 . . 3 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
32 eqid 2621 . . 3 (UnifSt‘(𝑊s 𝐴)) = (UnifSt‘(𝑊s 𝐴))
3319, 6resstopn 20900 . . 3 (𝐽t 𝐴) = (TopOpen‘(𝑊s 𝐴))
3431, 32, 33isusp 21975 . 2 ((𝑊s 𝐴) ∈ UnifSp ↔ ((UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))) ∧ (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴)))))
3523, 30, 34sylanbrc 697 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3555   × cxp 5072  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  t crest 16002  TopOpenctopn 16003  TopOnctopon 20618  TopSpctps 20619  UnifOncust 21913  unifTopcutop 21944  UnifStcuss 21967  UnifSpcusp 21968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-tset 15881  df-unif 15886  df-rest 16004  df-topn 16005  df-top 20621  df-topon 20623  df-topsp 20624  df-ust 21914  df-utop 21945  df-uss 21970  df-usp 21971
This theorem is referenced by: (None)
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