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Theorem txlly 20643
 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 20479 . . 3 Locally
2 llytop 20479 . . 3 Locally
3 txtop 20576 . . 3
41, 2, 3syl2an 480 . 2 Locally Locally
5 eltx 20575 . . . 4 Locally Locally
6 simpll 759 . . . . . . . . 9 Locally Locally Locally
7 simprll 771 . . . . . . . . 9 Locally Locally
8 simprrl 773 . . . . . . . . . 10 Locally Locally
9 xp1st 6835 . . . . . . . . . 10
108, 9syl 17 . . . . . . . . 9 Locally Locally
11 llyi 20481 . . . . . . . . 9 Locally t
126, 7, 10, 11syl3anc 1265 . . . . . . . 8 Locally Locally t
13 simplr 761 . . . . . . . . 9 Locally Locally Locally
14 simprlr 772 . . . . . . . . 9 Locally Locally
15 xp2nd 6836 . . . . . . . . . 10
168, 15syl 17 . . . . . . . . 9 Locally Locally
17 llyi 20481 . . . . . . . . 9 Locally t
1813, 14, 16, 17syl3anc 1265 . . . . . . . 8 Locally Locally t
19 reeanv 2997 . . . . . . . . 9 t t t t
201ad3antrrr 735 . . . . . . . . . . . . . 14 Locally Locally t t
212ad3antlr 736 . . . . . . . . . . . . . 14 Locally Locally t t
22 simprll 771 . . . . . . . . . . . . . 14 Locally Locally t t
23 simprlr 772 . . . . . . . . . . . . . 14 Locally Locally t t
24 txopn 20609 . . . . . . . . . . . . . 14
2520, 21, 22, 23, 24syl22anc 1266 . . . . . . . . . . . . 13 Locally Locally t t
26 simprl1 1051 . . . . . . . . . . . . . . . 16 t t
27 simprr1 1054 . . . . . . . . . . . . . . . 16 t t
28 xpss12 4957 . . . . . . . . . . . . . . . 16
2926, 27, 28syl2anc 666 . . . . . . . . . . . . . . 15 t t
30 simprrr 774 . . . . . . . . . . . . . . 15 Locally Locally
3129, 30sylan9ssr 3479 . . . . . . . . . . . . . 14 Locally Locally t t
32 vex 3085 . . . . . . . . . . . . . . 15
3332elpw2 4586 . . . . . . . . . . . . . 14
3431, 33sylibr 216 . . . . . . . . . . . . 13 Locally Locally t t
3525, 34elind 3651 . . . . . . . . . . . 12 Locally Locally t t
36 1st2nd2 6842 . . . . . . . . . . . . . . 15
378, 36syl 17 . . . . . . . . . . . . . 14 Locally Locally
3837adantr 467 . . . . . . . . . . . . 13 Locally Locally t t
39 simprl2 1052 . . . . . . . . . . . . . . 15 t t
40 simprr2 1055 . . . . . . . . . . . . . . 15 t t
41 opelxpi 4883 . . . . . . . . . . . . . . 15
4239, 40, 41syl2anc 666 . . . . . . . . . . . . . 14 t t
4342adantl 468 . . . . . . . . . . . . 13 Locally Locally t t
4438, 43eqeltrd 2511 . . . . . . . . . . . 12 Locally Locally t t
45 txrest 20638 . . . . . . . . . . . . . 14 t t t
4620, 21, 22, 23, 45syl22anc 1266 . . . . . . . . . . . . 13 Locally Locally t t t t t
47 simprl3 1053 . . . . . . . . . . . . . . 15 t t t
48 simprr3 1056 . . . . . . . . . . . . . . 15 t t t
49 txlly.1 . . . . . . . . . . . . . . . 16
5049caovcl 6475 . . . . . . . . . . . . . . 15 t t t t
5147, 48, 50syl2anc 666 . . . . . . . . . . . . . 14 t t t t
5251adantl 468 . . . . . . . . . . . . 13 Locally Locally t t t t
5346, 52eqeltrd 2511 . . . . . . . . . . . 12 Locally Locally t t t
54 eleq2 2496 . . . . . . . . . . . . . 14
55 oveq2 6311 . . . . . . . . . . . . . . 15 t t
5655eleq1d 2492 . . . . . . . . . . . . . 14 t t
5754, 56anbi12d 716 . . . . . . . . . . . . 13 t t
5857rspcev 3183 . . . . . . . . . . . 12 t t
5935, 44, 53, 58syl12anc 1263 . . . . . . . . . . 11 Locally Locally t t t
6059expr 619 . . . . . . . . . 10 Locally Locally t t t
6160rexlimdvva 2925 . . . . . . . . 9 Locally Locally t t t
6219, 61syl5bir 222 . . . . . . . 8 Locally Locally t t t
6312, 18, 62mp2and 684 . . . . . . 7 Locally Locally t
6463expr 619 . . . . . 6 Locally Locally t
6564rexlimdvva 2925 . . . . 5 Locally Locally t
6665ralimdv 2836 . . . 4 Locally Locally t
675, 66sylbid 219 . . 3 Locally Locally t
6867ralrimiv 2838 . 2 Locally Locally t
69 islly 20475 . 2 Locally t
704, 68, 69sylanbrc 669 1 Locally Locally Locally
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   w3a 983   wceq 1438   wcel 1869  wral 2776  wrex 2777   cin 3436   wss 3437  cpw 3980  cop 4003   cxp 4849  cfv 5599  (class class class)co 6303  c1st 6803  c2nd 6804   ↾t crest 15312  ctop 19909  Locally clly 20471   ctx 20567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-rest 15314  df-topgen 15335  df-top 19913  df-bases 19914  df-lly 20473  df-tx 20569 This theorem is referenced by: (None)
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