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Theorem nllyidm 17557
 Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally " is a local property. (Use loclly 17555 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 17540 . . . 4 Locally 𝑛Locally
2 llyi 17542 . . . . . . 7 Locally 𝑛Locally t 𝑛Locally
3 simprr3 1008 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t 𝑛Locally
4 simprl 734 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally
5 ssid 3369 . . . . . . . . . . 11
65a1i 11 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally
7 simpl1 961 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally Locally 𝑛Locally
87, 1syl 16 . . . . . . . . . . 11 Locally 𝑛Locally t 𝑛Locally
9 restopn2 17246 . . . . . . . . . . 11 t
108, 4, 9syl2anc 644 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally t
114, 6, 10mpbir2and 890 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t
12 simprr2 1007 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally
13 nlly2i 17544 . . . . . . . . 9 t 𝑛Locally t t t t
143, 11, 12, 13syl3anc 1185 . . . . . . . 8 Locally 𝑛Locally t 𝑛Locally t t t
15 restopn2 17246 . . . . . . . . . . . . . 14 t
168, 4, 15syl2anc 644 . . . . . . . . . . . . 13 Locally 𝑛Locally t 𝑛Locally t
1716adantr 453 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally t
188adantr 453 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
19 simpr2l 1017 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
20 simpr31 1048 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
21 opnneip 17188 . . . . . . . . . . . . . . . . . 18
2218, 19, 20, 21syl3anc 1185 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
23 simpr32 1049 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
24 simpr1 964 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally t t
2524elpwid 3810 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
264adantr 453 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally t t
27 elssuni 4045 . . . . . . . . . . . . . . . . . . 19
2826, 27syl 16 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
2925, 28sstrd 3360 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
30 eqid 2438 . . . . . . . . . . . . . . . . . 18
3130ssnei2 17185 . . . . . . . . . . . . . . . . 17
3218, 22, 23, 29, 31syl22anc 1186 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t
33 simprr1 1006 . . . . . . . . . . . . . . . . . . 19 Locally 𝑛Locally t 𝑛Locally
3433adantr 453 . . . . . . . . . . . . . . . . . 18 Locally 𝑛Locally t 𝑛Locally t t
3525, 34sstrd 3360 . . . . . . . . . . . . . . . . 17 Locally 𝑛Locally t 𝑛Locally t t
36 vex 2961 . . . . . . . . . . . . . . . . . 18
3736elpw 3807 . . . . . . . . . . . . . . . . 17
3835, 37sylibr 205 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t
39 elin 3532 . . . . . . . . . . . . . . . 16
4032, 38, 39sylanbrc 647 . . . . . . . . . . . . . . 15 Locally 𝑛Locally t 𝑛Locally t t
41 restabs 17234 . . . . . . . . . . . . . . . . 17 t t t
4218, 25, 26, 41syl3anc 1185 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t t t t
43 simpr33 1050 . . . . . . . . . . . . . . . 16 Locally 𝑛Locally t 𝑛Locally t t t t
4442, 43eqeltrrd 2513 . . . . . . . . . . . . . . 15 Locally 𝑛Locally t 𝑛Locally t t t
4540, 44jca 520 . . . . . . . . . . . . . 14 Locally 𝑛Locally t 𝑛Locally t t t
46453exp2 1172 . . . . . . . . . . . . 13 Locally 𝑛Locally t 𝑛Locally t t t
4746imp 420 . . . . . . . . . . . 12 Locally 𝑛Locally t 𝑛Locally t t t
4817, 47sylbid 208 . . . . . . . . . . 11 Locally 𝑛Locally t 𝑛Locally t t t t
4948rexlimdv 2831 . . . . . . . . . 10 Locally 𝑛Locally t 𝑛Locally t t t t
5049expimpd 588 . . . . . . . . 9 Locally 𝑛Locally t 𝑛Locally t t t t
5150reximdv2 2817 . . . . . . . 8 Locally 𝑛Locally t 𝑛Locally t t t t
5214, 51mpd 15 . . . . . . 7 Locally 𝑛Locally t 𝑛Locally t
532, 52rexlimddv 2836 . . . . . 6 Locally 𝑛Locally t
54533expb 1155 . . . . 5 Locally 𝑛Locally t
5554ralrimivva 2800 . . . 4 Locally 𝑛Locally t
56 isnlly 17537 . . . 4 𝑛Locally t
571, 55, 56sylanbrc 647 . . 3 Locally 𝑛Locally 𝑛Locally
5857ssriv 3354 . 2 Locally 𝑛Locally 𝑛Locally
59 nllyrest 17554 . . . . 5 𝑛Locally t 𝑛Locally
6059adantl 454 . . . 4 𝑛Locally t 𝑛Locally
61 nllytop 17541 . . . . . 6 𝑛Locally
6261ssriv 3354 . . . . 5 𝑛Locally
6362a1i 11 . . . 4 𝑛Locally
6460, 63restlly 17551 . . 3 𝑛Locally Locally 𝑛Locally
6564trud 1333 . 2 𝑛Locally Locally 𝑛Locally
6658, 65eqssi 3366 1 Locally 𝑛Locally 𝑛Locally
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wtru 1326   wceq 1653   wcel 1726  wral 2707  wrex 2708   cin 3321   wss 3322  cpw 3801  csn 3816  cuni 4017  cfv 5457  (class class class)co 6084   ↾t crest 13653  ctop 16963  cnei 17166  Locally clly 17532  𝑛Locally cnlly 17533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-nei 17167  df-lly 17534  df-nlly 17535
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