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Theorem ressusp 22881
 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 22879 . . . . 5 (𝐴𝐽 → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
213ad2ant3 1132 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
3 simp1 1133 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ UnifSp)
4 ressusp.1 . . . . . . . 8 𝐵 = (Base‘𝑊)
5 eqid 2798 . . . . . . . 8 (UnifSt‘𝑊) = (UnifSt‘𝑊)
6 ressusp.2 . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
74, 5, 6isusp 22877 . . . . . . 7 (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
83, 7sylib 221 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
98simpld 498 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘𝑊) ∈ (UnifOn‘𝐵))
10 simp2 1134 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ TopSp)
114, 6istps 21549 . . . . . . 7 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
1210, 11sylib 221 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 ∈ (TopOn‘𝐵))
13 simp3 1135 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐽)
14 toponss 21542 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐴𝐽) → 𝐴𝐵)
1512, 13, 14syl2anc 587 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐵)
16 trust 22845 . . . . 5 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴𝐵) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
179, 15, 16syl2anc 587 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
182, 17eqeltrd 2890 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘𝐴))
19 eqid 2798 . . . . . 6 (𝑊s 𝐴) = (𝑊s 𝐴)
2019, 4ressbas2 16550 . . . . 5 (𝐴𝐵𝐴 = (Base‘(𝑊s 𝐴)))
2115, 20syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 = (Base‘(𝑊s 𝐴)))
2221fveq2d 6650 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifOn‘𝐴) = (UnifOn‘(Base‘(𝑊s 𝐴))))
2318, 22eleqtrd 2892 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))))
248simprd 499 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 = (unifTop‘(UnifSt‘𝑊)))
2513, 24eleqtrd 2892 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 ∈ (unifTop‘(UnifSt‘𝑊)))
26 restutopopn 22854 . . . 4 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴 ∈ (unifTop‘(UnifSt‘𝑊))) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
279, 25, 26syl2anc 587 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
2824oveq1d 7151 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴))
292fveq2d 6650 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (unifTop‘(UnifSt‘(𝑊s 𝐴))) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
3027, 28, 293eqtr4d 2843 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴))))
31 eqid 2798 . . 3 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
32 eqid 2798 . . 3 (UnifSt‘(𝑊s 𝐴)) = (UnifSt‘(𝑊s 𝐴))
3319, 6resstopn 21801 . . 3 (𝐽t 𝐴) = (TopOpen‘(𝑊s 𝐴))
3431, 32, 33isusp 22877 . 2 ((𝑊s 𝐴) ∈ UnifSp ↔ ((UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))) ∧ (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴)))))
3523, 30, 34sylanbrc 586 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881   × cxp 5518  ‘cfv 6325  (class class class)co 7136  Basecbs 16478   ↾s cress 16479   ↾t crest 16689  TopOpenctopn 16690  TopOnctopon 21525  TopSpctps 21547  UnifOncust 22815  unifTopcutop 22846  UnifStcuss 22869  UnifSpcusp 22870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-tset 16579  df-unif 16583  df-rest 16691  df-topn 16692  df-top 21509  df-topon 21526  df-topsp 21548  df-ust 22816  df-utop 22847  df-uss 22872  df-usp 22873 This theorem is referenced by: (None)
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