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Theorem ressusp 23616
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 23614 . . . . 5 (𝐴𝐽 → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
213ad2ant3 1135 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
3 simp1 1136 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ UnifSp)
4 ressusp.1 . . . . . . . 8 𝐵 = (Base‘𝑊)
5 eqid 2736 . . . . . . . 8 (UnifSt‘𝑊) = (UnifSt‘𝑊)
6 ressusp.2 . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
74, 5, 6isusp 23613 . . . . . . 7 (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
83, 7sylib 217 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊))))
98simpld 495 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘𝑊) ∈ (UnifOn‘𝐵))
10 simp2 1137 . . . . . . 7 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝑊 ∈ TopSp)
114, 6istps 22283 . . . . . . 7 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
1210, 11sylib 217 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 ∈ (TopOn‘𝐵))
13 simp3 1138 . . . . . 6 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐽)
14 toponss 22276 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐴𝐽) → 𝐴𝐵)
1512, 13, 14syl2anc 584 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴𝐵)
16 trust 23581 . . . . 5 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴𝐵) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
179, 15, 16syl2anc 584 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
182, 17eqeltrd 2838 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘𝐴))
19 eqid 2736 . . . . . 6 (𝑊s 𝐴) = (𝑊s 𝐴)
2019, 4ressbas2 17120 . . . . 5 (𝐴𝐵𝐴 = (Base‘(𝑊s 𝐴)))
2115, 20syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 = (Base‘(𝑊s 𝐴)))
2221fveq2d 6846 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifOn‘𝐴) = (UnifOn‘(Base‘(𝑊s 𝐴))))
2318, 22eleqtrd 2840 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))))
248simprd 496 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐽 = (unifTop‘(UnifSt‘𝑊)))
2513, 24eleqtrd 2840 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → 𝐴 ∈ (unifTop‘(UnifSt‘𝑊)))
26 restutopopn 23590 . . . 4 (((UnifSt‘𝑊) ∈ (UnifOn‘𝐵) ∧ 𝐴 ∈ (unifTop‘(UnifSt‘𝑊))) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
279, 25, 26syl2anc 584 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
2824oveq1d 7372 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = ((unifTop‘(UnifSt‘𝑊)) ↾t 𝐴))
292fveq2d 6846 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (unifTop‘(UnifSt‘(𝑊s 𝐴))) = (unifTop‘((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))))
3027, 28, 293eqtr4d 2786 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴))))
31 eqid 2736 . . 3 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
32 eqid 2736 . . 3 (UnifSt‘(𝑊s 𝐴)) = (UnifSt‘(𝑊s 𝐴))
3319, 6resstopn 22537 . . 3 (𝐽t 𝐴) = (TopOpen‘(𝑊s 𝐴))
3431, 32, 33isusp 23613 . 2 ((𝑊s 𝐴) ∈ UnifSp ↔ ((UnifSt‘(𝑊s 𝐴)) ∈ (UnifOn‘(Base‘(𝑊s 𝐴))) ∧ (𝐽t 𝐴) = (unifTop‘(UnifSt‘(𝑊s 𝐴)))))
3523, 30, 34sylanbrc 583 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wss 3910   × cxp 5631  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  t crest 17302  TopOpenctopn 17303  TopOnctopon 22259  TopSpctps 22281  UnifOncust 23551  unifTopcutop 23582  UnifStcuss 23605  UnifSpcusp 23606
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-tset 17152  df-unif 17156  df-rest 17304  df-topn 17305  df-top 22243  df-topon 22260  df-topsp 22282  df-ust 23552  df-utop 23583  df-uss 23608  df-usp 23609
This theorem is referenced by: (None)
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