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Theorem mdslj2i 28054
Description: Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslle1.1  |-  A  e. 
CH
mdslle1.2  |-  B  e. 
CH
mdslle1.3  |-  C  e. 
CH
mdslle1.4  |-  D  e. 
CH
Assertion
Ref Expression
mdslj2i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  B
) )  ->  (
( C  i^i  D
)  vH  A )  =  ( ( C  vH  A )  i^i  ( D  vH  A
) ) )

Proof of Theorem mdslj2i
StepHypRef Expression
1 mdslle1.3 . . . 4  |-  C  e. 
CH
2 mdslle1.4 . . . 4  |-  D  e. 
CH
3 mdslle1.1 . . . 4  |-  A  e. 
CH
41, 2, 3lejdiri 27273 . . 3  |-  ( ( C  i^i  D )  vH  A )  C_  ( ( C  vH  A )  i^i  ( D  vH  A ) )
54a1i 11 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  B
) )  ->  (
( C  i^i  D
)  vH  A )  C_  ( ( C  vH  A )  i^i  ( D  vH  A ) ) )
6 ssin 3645 . . . . 5  |-  ( ( ( A  i^i  B
)  C_  C  /\  ( A  i^i  B ) 
C_  D )  <->  ( A  i^i  B )  C_  ( C  i^i  D ) )
76bicomi 207 . . . 4  |-  ( ( A  i^i  B ) 
C_  ( C  i^i  D )  <->  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )
)
8 mdslle1.2 . . . . . 6  |-  B  e. 
CH
91, 2, 8chlubi 27205 . . . . 5  |-  ( ( C  C_  B  /\  D  C_  B )  <->  ( C  vH  D )  C_  B
)
109bicomi 207 . . . 4  |-  ( ( C  vH  D ) 
C_  B  <->  ( C  C_  B  /\  D  C_  B ) )
117, 10anbi12i 711 . . 3  |-  ( ( ( A  i^i  B
)  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  B )  <->  ( (
( A  i^i  B
)  C_  C  /\  ( A  i^i  B ) 
C_  D )  /\  ( C  C_  B  /\  D  C_  B ) ) )
12 simpr 468 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
133, 1chub2i 27204 . . . . . . . 8  |-  A  C_  ( C  vH  A )
143, 2chub2i 27204 . . . . . . . 8  |-  A  C_  ( D  vH  A )
1513, 14ssini 3646 . . . . . . 7  |-  A  C_  ( ( C  vH  A )  i^i  ( D  vH  A ) )
1615a1i 11 . . . . . 6  |-  ( ( ( A  i^i  B
)  C_  C  /\  ( A  i^i  B ) 
C_  D )  ->  A  C_  ( ( C  vH  A )  i^i  ( D  vH  A
) ) )
171, 8, 3chlej1i 27207 . . . . . . . . 9  |-  ( C 
C_  B  ->  ( C  vH  A )  C_  ( B  vH  A ) )
188, 3chjcomi 27202 . . . . . . . . 9  |-  ( B  vH  A )  =  ( A  vH  B
)
1917, 18syl6sseq 3464 . . . . . . . 8  |-  ( C 
C_  B  ->  ( C  vH  A )  C_  ( A  vH  B ) )
20 ssinss1 3651 . . . . . . . 8  |-  ( ( C  vH  A ) 
C_  ( A  vH  B )  ->  (
( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( A  vH  B ) )
2119, 20syl 17 . . . . . . 7  |-  ( C 
C_  B  ->  (
( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( A  vH  B ) )
2221adantr 472 . . . . . 6  |-  ( ( C  C_  B  /\  D  C_  B )  -> 
( ( C  vH  A )  i^i  ( D  vH  A ) ) 
C_  ( A  vH  B ) )
231, 3chjcli 27191 . . . . . . . . 9  |-  ( C  vH  A )  e. 
CH
242, 3chjcli 27191 . . . . . . . . 9  |-  ( D  vH  A )  e. 
CH
2523, 24chincli 27194 . . . . . . . 8  |-  ( ( C  vH  A )  i^i  ( D  vH  A ) )  e. 
CH
263, 8, 253pm3.2i 1208 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  vH  A )  i^i  ( D  vH  A ) )  e. 
CH )
27 dmdsl3 28049 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  vH  A )  i^i  ( D  vH  A ) )  e.  CH )  /\  ( B  MH*  A  /\  A  C_  ( ( C  vH  A )  i^i  ( D  vH  A
) )  /\  (
( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( A  vH  B ) ) )  ->  (
( ( ( C  vH  A )  i^i  ( D  vH  A
) )  i^i  B
)  vH  A )  =  ( ( C  vH  A )  i^i  ( D  vH  A
) ) )
2826, 27mpan 684 . . . . . 6  |-  ( ( B  MH*  A  /\  A  C_  ( ( C  vH  A )  i^i  ( D  vH  A
) )  /\  (
( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( A  vH  B ) )  ->  ( (
( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i  B )  vH  A )  =  ( ( C  vH  A
)  i^i  ( D  vH  A ) ) )
2912, 16, 22, 28syl3an 1334 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( (
( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i  B )  vH  A )  =  ( ( C  vH  A
)  i^i  ( D  vH  A ) ) )
30 inss1 3643 . . . . . . . . 9  |-  ( ( C  vH  A )  i^i  ( D  vH  A ) )  C_  ( C  vH  A )
31 ssrin 3648 . . . . . . . . 9  |-  ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( C  vH  A )  ->  ( ( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i 
B )  C_  (
( C  vH  A
)  i^i  B )
)
3230, 31ax-mp 5 . . . . . . . 8  |-  ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  C_  (
( C  vH  A
)  i^i  B )
33 simpl 464 . . . . . . . . 9  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
34 simpl 464 . . . . . . . . 9  |-  ( ( ( A  i^i  B
)  C_  C  /\  ( A  i^i  B ) 
C_  D )  -> 
( A  i^i  B
)  C_  C )
35 simpl 464 . . . . . . . . 9  |-  ( ( C  C_  B  /\  D  C_  B )  ->  C  C_  B )
363, 8, 13pm3.2i 1208 . . . . . . . . . 10  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
37 mdsl3 28050 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  C  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  C )
3836, 37mpan 684 . . . . . . . . 9  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  C  /\  C  C_  B )  ->  (
( C  vH  A
)  i^i  B )  =  C )
3933, 34, 35, 38syl3an 1334 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  C )
4032, 39syl5sseq 3466 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( (
( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  C_  C
)
41 inss2 3644 . . . . . . . . 9  |-  ( ( C  vH  A )  i^i  ( D  vH  A ) )  C_  ( D  vH  A )
42 ssrin 3648 . . . . . . . . 9  |-  ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( D  vH  A )  ->  ( ( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i 
B )  C_  (
( D  vH  A
)  i^i  B )
)
4341, 42ax-mp 5 . . . . . . . 8  |-  ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  C_  (
( D  vH  A
)  i^i  B )
44 simpr 468 . . . . . . . . 9  |-  ( ( ( A  i^i  B
)  C_  C  /\  ( A  i^i  B ) 
C_  D )  -> 
( A  i^i  B
)  C_  D )
45 simpr 468 . . . . . . . . 9  |-  ( ( C  C_  B  /\  D  C_  B )  ->  D  C_  B )
463, 8, 23pm3.2i 1208 . . . . . . . . . 10  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
47 mdsl3 28050 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  D  /\  D  C_  B ) )  ->  ( ( D  vH  A )  i^i 
B )  =  D )
4846, 47mpan 684 . . . . . . . . 9  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  D  /\  D  C_  B )  ->  (
( D  vH  A
)  i^i  B )  =  D )
4933, 44, 45, 48syl3an 1334 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( ( D  vH  A )  i^i 
B )  =  D )
5043, 49syl5sseq 3466 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( (
( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  C_  D
)
5140, 50ssind 3647 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( (
( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  C_  ( C  i^i  D ) )
5225, 8chincli 27194 . . . . . . 7  |-  ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  e.  CH
531, 2chincli 27194 . . . . . . 7  |-  ( C  i^i  D )  e. 
CH
5452, 53, 3chlej1i 27207 . . . . . 6  |-  ( ( ( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i  B )  C_  ( C  i^i  D )  ->  ( ( ( ( C  vH  A
)  i^i  ( D  vH  A ) )  i^i 
B )  vH  A
)  C_  ( ( C  i^i  D )  vH  A ) )
5551, 54syl 17 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( (
( ( C  vH  A )  i^i  ( D  vH  A ) )  i^i  B )  vH  A )  C_  (
( C  i^i  D
)  vH  A )
)
5629, 55eqsstr3d 3453 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) )  ->  ( ( C  vH  A )  i^i  ( D  vH  A
) )  C_  (
( C  i^i  D
)  vH  A )
)
57563expb 1232 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( ( A  i^i  B ) 
C_  C  /\  ( A  i^i  B )  C_  D )  /\  ( C  C_  B  /\  D  C_  B ) ) )  ->  ( ( C  vH  A )  i^i  ( D  vH  A
) )  C_  (
( C  i^i  D
)  vH  A )
)
5811, 57sylan2b 483 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  B
) )  ->  (
( C  vH  A
)  i^i  ( D  vH  A ) )  C_  ( ( C  i^i  D )  vH  A ) )
595, 58eqssd 3435 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  B
) )  ->  (
( C  i^i  D
)  vH  A )  =  ( ( C  vH  A )  i^i  ( D  vH  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   class class class wbr 4395  (class class class)co 6308   CHcch 26663    vH chj 26667    MH cmd 26700    MH* cdmd 26701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637  ax-hilex 26733  ax-hfvadd 26734  ax-hvcom 26735  ax-hvass 26736  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvdistr1 26742  ax-hvdistr2 26743  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his2 26817  ax-his3 26818  ax-his4 26819  ax-hcompl 26936
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-lm 20322  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cfil 22303  df-cau 22304  df-cmet 22305  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-vs 26299  df-nmcv 26300  df-ims 26301  df-dip 26418  df-ssp 26442  df-ph 26535  df-cbn 26586  df-hnorm 26702  df-hba 26703  df-hvsub 26705  df-hlim 26706  df-hcau 26707  df-sh 26941  df-ch 26955  df-oc 26986  df-ch0 26987  df-shs 27042  df-chj 27044  df-md 28014  df-dmd 28015
This theorem is referenced by: (None)
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