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Theorem nllyidm 19863
 Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally " is a local property. (Use loclly 19861 to show 𝑛Locally 𝑛Locally 𝑛Locally .) (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyidm Locally 𝑛Locally 𝑛Locally

Proof of Theorem nllyidm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 19846 . . . 4 Locally 𝑛Locally
2 llyi 19848 . . . . . . 7 Locally 𝑛Locally t 𝑛Locally
3 simprr3 1047 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t 𝑛Locally
4 simprl 756 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally
5 ssid 3508 . . . . . . . . . . 11
65a1i 11 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally
7 simpl1 1000 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally Locally 𝑛Locally
87, 1syl 16 . . . . . . . . . . 11 Locally 𝑛Locally t 𝑛Locally
9 restopn2 19551 . . . . . . . . . . 11 t
108, 4, 9syl2anc 661 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally t
114, 6, 10mpbir2and 922 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t
12 simprr2 1046 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally
13 nlly2i 19850 . . . . . . . . 9 t 𝑛Locally t t t t
143, 11, 12, 13syl3anc 1229 . . . . . . . 8 Locally 𝑛Locally t 𝑛Locally t t t
15 restopn2 19551 . . . . . . . . . . . . . 14 t
168, 4, 15syl2anc 661 . . . . . . . . . . . . 13 Locally 𝑛Locally t 𝑛Locally t
1716adantr 465 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally t
188adantr 465 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
19 simpr2l 1056 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
20 simpr31 1087 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
21 opnneip 19493 . . . . . . . . . . . . . . . . . 18
2218, 19, 20, 21syl3anc 1229 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
23 simpr32 1088 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
24 simpr1 1003 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally t t
2524elpwid 4007 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
264adantr 465 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally t t
27 elssuni 4264 . . . . . . . . . . . . . . . . . . 19
2826, 27syl 16 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
2925, 28sstrd 3499 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
30 eqid 2443 . . . . . . . . . . . . . . . . . 18
3130ssnei2 19490 . . . . . . . . . . . . . . . . 17
3218, 22, 23, 29, 31syl22anc 1230 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t
33 simprr1 1045 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally
3433adantr 465 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
3525, 34sstrd 3499 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
36 selpw 4004 . . . . . . . . . . . . . . . . 17
3735, 36sylibr 212 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t
3832, 37elind 3673 . . . . . . . . . . . . . . 15 Locally 𝑛Locally t 𝑛Locally t t
39 restabs 19539 . . . . . . . . . . . . . . . . 17 t t t
4018, 25, 26, 39syl3anc 1229 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t t t t
41 simpr33 1089 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t t t
4240, 41eqeltrrd 2532 . . . . . . . . . . . . . . 15 Locally 𝑛Locally t 𝑛Locally t t t
4338, 42jca 532 . . . . . . . . . . . . . 14 Locally 𝑛Locally t 𝑛Locally t t t
44433exp2 1215 . . . . . . . . . . . . 13 Locally 𝑛Locally t 𝑛Locally t t t
4544imp 429 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally t t t
4617, 45sylbid 215 . . . . . . . . . . 11 Locally 𝑛Locally t 𝑛Locally t t t t
4746rexlimdv 2933 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally t t t t
4847expimpd 603 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t t t t
4948reximdv2 2914 . . . . . . . 8 Locally 𝑛Locally t 𝑛Locally t t t t
5014, 49mpd 15 . . . . . . 7 Locally 𝑛Locally t 𝑛Locally t
512, 50rexlimddv 2939 . . . . . 6 Locally 𝑛Locally t
52513expb 1198 . . . . 5 Locally 𝑛Locally t
5352ralrimivva 2864 . . . 4 Locally 𝑛Locally t
54 isnlly 19843 . . . 4 𝑛Locally t
551, 53, 54sylanbrc 664 . . 3 Locally 𝑛Locally 𝑛Locally
5655ssriv 3493 . 2 Locally 𝑛Locally 𝑛Locally
57 nllyrest 19860 . . . . 5 𝑛Locally t 𝑛Locally
5857adantl 466 . . . 4 𝑛Locally t 𝑛Locally
59 nllytop 19847 . . . . . 6 𝑛Locally
6059ssriv 3493 . . . . 5 𝑛Locally
6160a1i 11 . . . 4 𝑛Locally
6258, 61restlly 19857 . . 3 𝑛Locally Locally 𝑛Locally
6362trud 1392 . 2 𝑛Locally Locally 𝑛Locally
6456, 63eqssi 3505 1 Locally 𝑛Locally 𝑛Locally
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 974   wceq 1383   wtru 1384   wcel 1804  wral 2793  wrex 2794   cin 3460   wss 3461  cpw 3997  csn 4014  cuni 4234  cfv 5578  (class class class)co 6281   ↾t crest 14695  ctop 19267  cnei 19471  Locally clly 19838  𝑛Locally cnlly 19839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-er 7313  df-en 7519  df-fin 7522  df-fi 7873  df-rest 14697  df-topgen 14718  df-top 19272  df-bases 19274  df-topon 19275  df-nei 19472  df-lly 19840  df-nlly 19841 This theorem is referenced by: (None)
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