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Theorem mdsl1i 25723
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 3376 . . . . . . . 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 3375 . . . . . . . 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 3375 . . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( x C_ B <-> ( y vH ( A i^i B ) ) C_ B ) )
5 oveq1 6096 . . . . . . . . . 10 |- ( x = ( y vH ( A i^i B ) ) -> ( x vH A ) = ( ( y vH ( A i^i B ) ) vH A ) )
65ineq1d 3549 . . . . . . . . 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 6096 . . . . . . . . 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 2455 . . . . . . . 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 3068 . . . . 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 3569 . . . . . . . . . . . 12 |- ( A i^i B ) C_ B
16 mdsl.1 . . . . . . . . . . . . . . 15 |- A e. CH
17 mdsl.2 . . . . . . . . . . . . . . 15 |- B e. CH
1816, 17chincli 24861 . . . . . . . . . . . . . 14 |- ( A i^i B ) e. CH
19 chlub 24910 . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . 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 24871 . . . . . . . . . . . 12 |- B C_ ( A vH B )
24 sstr 3362 . . . . . . . . . . . 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 24909 . . . . . . . . . . 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 24758 . . . . . . . . . 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 24934 . . . . . . . . . . . 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 1306 . . . . . . . . . . 11 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH ( ( A i^i B ) vH A ) ) )
3618, 16chjcomi 24869 . . . . . . . . . . . . 13 |- ( ( A i^i B ) vH A ) = ( A vH ( A i^i B ) )
3716, 17chabs1i 24919 . . . . . . . . . . . . 13 |- ( A vH ( A i^i B ) ) = A
3836, 37eqtri 2461 . . . . . . . . . . . 12 |- ( ( A i^i B ) vH A ) = A
3938oveq2i 6100 . . . . . . . . . . 11 |- ( y vH ( ( A i^i B ) vH A ) ) = ( y vH A )
4035, 39syl6eq 2489 . . . . . . . . . 10 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH A ) )
4140ineq1d 3549 . . . . . . . . 9 |- ( y e. CH -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH A ) i^i B ) )
42 chjass 24934 . . . . . . . . . . 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 1306 . . . . . . . . . 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 24922 . . . . . . . . . . 11 |- ( ( A i^i B ) vH ( A i^i B ) ) = ( A i^i B )
4544oveq2i 6100 . . . . . . . . . 10 |- ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) = ( y vH ( A i^i B ) )
4643, 45syl6eq 2489 . . . . . . . . 9 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( A i^i B ) ) )
4741, 46eqeq12d 2455 . . . . . . . 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 2796 . . 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 25696 . . . 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 25696 . . . 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 2789 . . 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 1369   e. wcel 1756   A.wral 2713   i^i cin 3325   C_ wss 3326   class class class wbr 4290  (class class class)co 6089   CHcch 24329   vH chj 24333   MH cmd 24366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cc 8602  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360  ax-hilex 24399  ax-hfvadd 24400  ax-hvcom 24401  ax-hvass 24402  ax-hv0cl 24403  ax-hvaddid 24404  ax-hfvmul 24405  ax-hvmulid 24406  ax-hvmulass 24407  ax-hvdistr1 24408  ax-hvdistr2 24409  ax-hvmul0 24410  ax-hfi 24479  ax-his1 24482  ax-his2 24483  ax-his3 24484  ax-his4 24485  ax-hcompl 24602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-omul 6923  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-acn 8110  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-fl 11640  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-rlim 12965  df-sum 13162  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-submnd 15463  df-mulg 15546  df-cntz 15833  df-cmn 16277  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-fbas 17812  df-fg 17813  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cld 18621  df-ntr 18622  df-cls 18623  df-nei 18700  df-cn 18829  df-cnp 18830  df-lm 18831  df-haus 18917  df-tx 19133  df-hmeo 19326  df-fil 19417  df-fm 19509  df-flim 19510  df-flf 19511  df-xms 19893  df-ms 19894  df-tms 19895  df-cfil 20764  df-cau 20765  df-cmet 20766  df-grpo 23676  df-gid 23677  df-ginv 23678  df-gdiv 23679  df-ablo 23767  df-subgo 23787  df-vc 23922  df-nv 23968  df-va 23971  df-ba 23972  df-sm 23973  df-0v 23974  df-vs 23975  df-nmcv 23976  df-ims 23977  df-dip 24094  df-ssp 24118  df-ph 24211  df-cbn 24262  df-hnorm 24368  df-hba 24369  df-hvsub 24371  df-hlim 24372  df-hcau 24373  df-sh 24607  df-ch 24622  df-oc 24653  df-ch0 24654  df-shs 24709  df-chj 24711  df-md 25682
This theorem is referenced by:  mdsl2i  25724  cvmdi  25726
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