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Theorem mdsl1i 27366
Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdsl.1 |- A e. CH
mdsl.2 |- B e. CH
Assertion
Ref Expression
mdsl1i |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> A MH B )
Distinct variable groups:   x, A   x, B
Proof of Theorem mdsl1i
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sseq2 3521 . . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( ( A i^i B ) C_ x <-> ( A i^i B ) C_ ( y vH ( A i^i B ) ) ) )
2 sseq1 3520 . . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( x C_ ( A vH B ) <-> ( y vH ( A i^i B ) ) C_ ( A vH B ) ) )
31, 2anbi12d 710 . . . . . . 7 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) <-> ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) )
4 sseq1 3520 . . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( x C_ B <-> ( y vH ( A i^i B ) ) C_ B ) )
5 oveq1 6303 . . . . . . . . . 10 |- ( x = ( y vH ( A i^i B ) ) -> ( x vH A ) = ( ( y vH ( A i^i B ) ) vH A ) )
65ineq1d 3695 . . . . . . . . 9 |- ( x = ( y vH ( A i^i B ) ) -> ( ( x vH A ) i^i B ) = ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) )
7 oveq1 6303 . . . . . . . . 9 |- ( x = ( y vH ( A i^i B ) ) -> ( x vH ( A i^i B ) ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) )
86, 7eqeq12d 2479 . . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) <-> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) )
94, 8imbi12d 320 . . . . . . 7 |- ( x = ( y vH ( A i^i B ) ) -> ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) )
103, 9imbi12d 320 . . . . . 6 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
1110rspccv 3207 . . . . 5 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
12 impexp 446 . . . . . . 7 |- ( ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) <-> ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) )
13 impexp 446 . . . . . . 7 |- ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) <-> ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
1412, 13bitr2i 250 . . . . . 6 |- ( ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) <-> ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) )
15 inss2 3715 . . . . . . . . . . . 12 |- ( A i^i B ) C_ B
16 mdsl.1 . . . . . . . . . . . . . . 15 |- A e. CH
17 mdsl.2 . . . . . . . . . . . . . . 15 |- B e. CH
1816, 17chincli 26504 . . . . . . . . . . . . . 14 |- ( A i^i B ) e. CH
19 chlub 26553 . . . . . . . . . . . . . 14 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ B e. CH ) -> ( ( y C_ B /\ ( A i^i B ) C_ B ) <-> ( y vH ( A i^i B ) ) C_ B ) )
2018, 17, 19mp3an23 1316 . . . . . . . . . . . . 13 |- ( y e. CH -> ( ( y C_ B /\ ( A i^i B ) C_ B ) <-> ( y vH ( A i^i B ) ) C_ B ) )
2120biimpd 207 . . . . . . . . . . . 12 |- ( y e. CH -> ( ( y C_ B /\ ( A i^i B ) C_ B ) -> ( y vH ( A i^i B ) ) C_ B ) )
2215, 21mpan2i 677 . . . . . . . . . . 11 |- ( y e. CH -> ( y C_ B -> ( y vH ( A i^i B ) ) C_ B ) )
2317, 16chub2i 26514 . . . . . . . . . . . 12 |- B C_ ( A vH B )
24 sstr 3507 . . . . . . . . . . . 12 |- ( ( ( y vH ( A i^i B ) ) C_ B /\ B C_ ( A vH B ) ) -> ( y vH ( A i^i B ) ) C_ ( A vH B ) )
2523, 24mpan2 671 . . . . . . . . . . 11 |- ( ( y vH ( A i^i B ) ) C_ B -> ( y vH ( A i^i B ) ) C_ ( A vH B ) )
2622, 25syl6 33 . . . . . . . . . 10 |- ( y e. CH -> ( y C_ B -> ( y vH ( A i^i B ) ) C_ ( A vH B ) ) )
27 chub2 26552 . . . . . . . . . . 11 |- ( ( ( A i^i B ) e. CH /\ y e. CH ) -> ( A i^i B ) C_ ( y vH ( A i^i B ) ) )
2818, 27mpan 670 . . . . . . . . . 10 |- ( y e. CH -> ( A i^i B ) C_ ( y vH ( A i^i B ) ) )
2926, 28jctild 543 . . . . . . . . 9 |- ( y e. CH -> ( y C_ B -> ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) )
30 chjcl 26401 . . . . . . . . . 10 |- ( ( y e. CH /\ ( A i^i B ) e. CH ) -> ( y vH ( A i^i B ) ) e. CH )
3118, 30mpan2 671 . . . . . . . . 9 |- ( y e. CH -> ( y vH ( A i^i B ) ) e. CH )
3229, 31jctild 543 . . . . . . . 8 |- ( y e. CH -> ( y C_ B -> ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) ) )
3332, 22jcad 533 . . . . . . 7 |- ( y e. CH -> ( y C_ B -> ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) ) )
34 chjass 26577 . . . . . . . . . . . 12 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ A e. CH ) -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH ( ( A i^i B ) vH A ) ) )
3518, 16, 34mp3an23 1316 . . . . . . . . . . 11 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH ( ( A i^i B ) vH A ) ) )
3618, 16chjcomi 26512 . . . . . . . . . . . . 13 |- ( ( A i^i B ) vH A ) = ( A vH ( A i^i B ) )
3716, 17chabs1i 26562 . . . . . . . . . . . . 13 |- ( A vH ( A i^i B ) ) = A
3836, 37eqtri 2486 . . . . . . . . . . . 12 |- ( ( A i^i B ) vH A ) = A
3938oveq2i 6307 . . . . . . . . . . 11 |- ( y vH ( ( A i^i B ) vH A ) ) = ( y vH A )
4035, 39syl6eq 2514 . . . . . . . . . 10 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH A ) )
4140ineq1d 3695 . . . . . . . . 9 |- ( y e. CH -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH A ) i^i B ) )
42 chjass 26577 . . . . . . . . . . 11 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ ( A i^i B ) e. CH ) -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) )
4318, 18, 42mp3an23 1316 . . . . . . . . . 10 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) )
4418chjidmi 26565 . . . . . . . . . . 11 |- ( ( A i^i B ) vH ( A i^i B ) ) = ( A i^i B )
4544oveq2i 6307 . . . . . . . . . 10 |- ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) = ( y vH ( A i^i B ) )
4643, 45syl6eq 2514 . . . . . . . . 9 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( A i^i B ) ) )
4741, 46eqeq12d 2479 . . . . . . . 8 |- ( y e. CH -> ( ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) <-> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
4847biimpd 207 . . . . . . 7 |- ( y e. CH -> ( ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
4933, 48imim12d 74 . . . . . 6 |- ( y e. CH -> ( ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
5014, 49syl5bi 217 . . . . 5 |- ( y e. CH -> ( ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
5111, 50syl5com 30 . . . 4 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> ( y e. CH -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
5251ralrimiv 2869 . . 3 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
53 mdbr 27339 . . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
5416, 17, 53mp2an 672 . . 3 |- ( A MH B <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
5552, 54sylibr 212 . 2 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> A MH B )
56 mdbr 27339 . . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
5716, 17, 56mp2an 672 . . 3 |- ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) )
58 ax-1 6 . . . 4 |- ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
5958ralimi 2850 . . 3 |- ( A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
6057, 59sylbi 195 . 2 |- ( A MH B -> A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
6155, 60impbii 188 1 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> A MH B )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   i^i cin 3470   C_ wss 3471   class class class wbr 4456  (class class class)co 6296   CHcch 25972   vH chj 25976   MH cmd 26009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589  ax-hilex 26042  ax-hfvadd 26043  ax-hvcom 26044  ax-hvass 26045  ax-hv0cl 26046  ax-hvaddid 26047  ax-hfvmul 26048  ax-hvmulid 26049  ax-hvmulass 26050  ax-hvdistr1 26051  ax-hvdistr2 26052  ax-hvmul0 26053  ax-hfi 26122  ax-his1 26125  ax-his2 26126  ax-his3 26127  ax-his4 26128  ax-hcompl 26245
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-rlim 13323  df-sum 13520  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-cn 19854  df-cnp 19855  df-lm 19856  df-haus 19942  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cfil 21819  df-cau 21820  df-cmet 21821  df-grpo 25319  df-gid 25320  df-ginv 25321  df-gdiv 25322  df-ablo 25410  df-subgo 25430  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-vs 25618  df-nmcv 25619  df-ims 25620  df-dip 25737  df-ssp 25761  df-ph 25854  df-cbn 25905  df-hnorm 26011  df-hba 26012  df-hvsub 26014  df-hlim 26015  df-hcau 26016  df-sh 26250  df-ch 26265  df-oc 26296  df-ch0 26297  df-shs 26352  df-chj 26354  df-md 27325
This theorem is referenced by:  mdsl2i  27367  cvmdi  27369
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