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Theorem txlly 17330
 Description: If the property is preserved under topological products, then so is the property of being locally . (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
txlly.1
Assertion
Ref Expression
txlly Locally Locally Locally
Distinct variable groups:   ,,   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem txlly
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 17198 . . 3 Locally
2 llytop 17198 . . 3 Locally
3 txtop 17264 . . 3
41, 2, 3syl2an 463 . 2 Locally Locally
5 eltx 17263 . . . 4 Locally Locally
6 simpll 730 . . . . . . . . 9 Locally Locally Locally
7 simprll 738 . . . . . . . . 9 Locally Locally
8 simprrl 740 . . . . . . . . . 10 Locally Locally
9 xp1st 6149 . . . . . . . . . 10
108, 9syl 15 . . . . . . . . 9 Locally Locally
11 llyi 17200 . . . . . . . . 9 Locally t
126, 7, 10, 11syl3anc 1182 . . . . . . . 8 Locally Locally t
13 simplr 731 . . . . . . . . 9 Locally Locally Locally
14 simprlr 739 . . . . . . . . 9 Locally Locally
15 xp2nd 6150 . . . . . . . . . 10
168, 15syl 15 . . . . . . . . 9 Locally Locally
17 llyi 17200 . . . . . . . . 9 Locally t
1813, 14, 16, 17syl3anc 1182 . . . . . . . 8 Locally Locally t
19 reeanv 2707 . . . . . . . . 9 t t t t
201ad3antrrr 710 . . . . . . . . . . . . . 14 Locally Locally t t
212ad2antlr 707 . . . . . . . . . . . . . . 15 Locally Locally
2221adantr 451 . . . . . . . . . . . . . 14 Locally Locally t t
23 simprll 738 . . . . . . . . . . . . . 14 Locally Locally t t
24 simprlr 739 . . . . . . . . . . . . . 14 Locally Locally t t
25 txopn 17297 . . . . . . . . . . . . . 14
2620, 22, 23, 24, 25syl22anc 1183 . . . . . . . . . . . . 13 Locally Locally t t
27 simprl1 1000 . . . . . . . . . . . . . . . 16 t t
28 simprr1 1003 . . . . . . . . . . . . . . . 16 t t
29 xpss12 4792 . . . . . . . . . . . . . . . 16
3027, 28, 29syl2anc 642 . . . . . . . . . . . . . . 15 t t
31 simprrr 741 . . . . . . . . . . . . . . 15 Locally Locally
3230, 31sylan9ssr 3193 . . . . . . . . . . . . . 14 Locally Locally t t
33 vex 2791 . . . . . . . . . . . . . . 15
3433elpw2 4175 . . . . . . . . . . . . . 14
3532, 34sylibr 203 . . . . . . . . . . . . 13 Locally Locally t t
36 elin 3358 . . . . . . . . . . . . 13
3726, 35, 36sylanbrc 645 . . . . . . . . . . . 12 Locally Locally t t
38 1st2nd2 6159 . . . . . . . . . . . . . . 15
398, 38syl 15 . . . . . . . . . . . . . 14 Locally Locally
4039adantr 451 . . . . . . . . . . . . 13 Locally Locally t t
41 simprl2 1001 . . . . . . . . . . . . . . 15 t t
42 simprr2 1004 . . . . . . . . . . . . . . 15 t t
43 opelxpi 4721 . . . . . . . . . . . . . . 15
4441, 42, 43syl2anc 642 . . . . . . . . . . . . . 14 t t
4544adantl 452 . . . . . . . . . . . . 13 Locally Locally t t
4640, 45eqeltrd 2357 . . . . . . . . . . . 12 Locally Locally t t
47 txrest 17325 . . . . . . . . . . . . . 14 t t t
4820, 22, 23, 24, 47syl22anc 1183 . . . . . . . . . . . . 13 Locally Locally t t t t t
49 simprl3 1002 . . . . . . . . . . . . . . 15 t t t
50 simprr3 1005 . . . . . . . . . . . . . . 15 t t t
51 txlly.1 . . . . . . . . . . . . . . . 16
5251caovcl 6014 . . . . . . . . . . . . . . 15 t t t t
5349, 50, 52syl2anc 642 . . . . . . . . . . . . . 14 t t t t
5453adantl 452 . . . . . . . . . . . . 13 Locally Locally t t t t
5548, 54eqeltrd 2357 . . . . . . . . . . . 12 Locally Locally t t t
56 eleq2 2344 . . . . . . . . . . . . . 14
57 oveq2 5866 . . . . . . . . . . . . . . 15 t t
5857eleq1d 2349 . . . . . . . . . . . . . 14 t t
5956, 58anbi12d 691 . . . . . . . . . . . . 13 t t
6059rspcev 2884 . . . . . . . . . . . 12 t t
6137, 46, 55, 60syl12anc 1180 . . . . . . . . . . 11 Locally Locally t t t
6261expr 598 . . . . . . . . . 10 Locally Locally t t t
6362rexlimdvva 2674 . . . . . . . . 9 Locally Locally t t t
6419, 63syl5bir 209 . . . . . . . 8 Locally Locally t t t
6512, 18, 64mp2and 660 . . . . . . 7 Locally Locally t
6665expr 598 . . . . . 6 Locally Locally t
6766rexlimdvva 2674 . . . . 5 Locally Locally t
6867ralimdv 2622 . . . 4 Locally Locally t
695, 68sylbid 206 . . 3 Locally Locally t
7069ralrimiv 2625 . 2 Locally Locally t
71 islly 17194 . 2 Locally t
724, 70, 71sylanbrc 645 1 Locally Locally Locally
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  wral 2543  wrex 2544   cin 3151   wss 3152  cpw 3625  cop 3643   cxp 4687  cfv 5255  (class class class)co 5858  c1st 6120  c2nd 6121   ↾t crest 13325  ctop 16631  Locally clly 17190   ctx 17255 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-lly 17192  df-tx 17257
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