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Theorem mdslj1i 25860
Description: Join preservation of the one-to-one onto 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
mdslj1i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )

Proof of Theorem mdslj1i
StepHypRef Expression
1 ssin 3672 . . . . 5  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
21bicomi 202 . . . 4  |-  ( A 
C_  ( C  i^i  D )  <->  ( A  C_  C  /\  A  C_  D
) )
3 mdslle1.3 . . . . . 6  |-  C  e. 
CH
4 mdslle1.4 . . . . . 6  |-  D  e. 
CH
5 mdslle1.1 . . . . . . 7  |-  A  e. 
CH
6 mdslle1.2 . . . . . . 7  |-  B  e. 
CH
75, 6chjcli 24997 . . . . . 6  |-  ( A  vH  B )  e. 
CH
83, 4, 7chlubi 25011 . . . . 5  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
98bicomi 202 . . . 4  |-  ( ( C  vH  D ) 
C_  ( A  vH  B )  <->  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )
102, 9anbi12i 697 . . 3  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  <->  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )
11 simpr 461 . . . . . . . . . 10  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
12 simpl 457 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  C )
13 simpl 457 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  C  C_  ( A  vH  B
) )
145, 6, 33pm3.2i 1166 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
15 dmdsl3 25856 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
1614, 15mpan 670 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  A )  =  C )
1711, 12, 13, 16syl3an 1261 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
183, 6chincli 25000 . . . . . . . . . . 11  |-  ( C  i^i  B )  e. 
CH
194, 6chincli 25000 . . . . . . . . . . 11  |-  ( D  i^i  B )  e. 
CH
2018, 19chub1i 25009 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
2118, 19chjcli 24997 . . . . . . . . . . 11  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH
2218, 21, 5chlej1i 25013 . . . . . . . . . 10  |-  ( ( C  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2320, 22mp1i 12 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2417, 23eqsstr3d 3491 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  C  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
25 simpr 461 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
26 simpr 461 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
275, 6, 43pm3.2i 1166 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
28 dmdsl3 25856 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
2927, 28mpan 670 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
3011, 25, 26, 29syl3an 1261 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
3119, 18chub2i 25010 . . . . . . . . . 10  |-  ( D  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
3219, 21, 5chlej1i 25013 . . . . . . . . . 10  |-  ( ( D  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3331, 32mp1i 12 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3430, 33eqsstr3d 3491 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3524, 34jca 532 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) ) )
3621, 5chjcli 24997 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  e.  CH
373, 4, 36chlubi 25011 . . . . . . 7  |-  ( ( C  C_  ( (
( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )  <->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3835, 37sylib 196 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
39 ssrin 3675 . . . . . 6  |-  ( ( C  vH  D ) 
C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A
)  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
4038, 39syl 16 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
41 simpl 457 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
42 ssrin 3675 . . . . . . . 8  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( C  i^i  B ) )
4342, 20syl6ss 3468 . . . . . . 7  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
4443adantr 465 . . . . . 6  |-  ( ( A  C_  C  /\  A  C_  D )  -> 
( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
45 inss2 3671 . . . . . . . 8  |-  ( C  i^i  B )  C_  B
46 inss2 3671 . . . . . . . 8  |-  ( D  i^i  B )  C_  B
4718, 19, 6chlubi 25011 . . . . . . . . 9  |-  ( ( ( C  i^i  B
)  C_  B  /\  ( D  i^i  B ) 
C_  B )  <->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B )
4847bicomi 202 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B 
<->  ( ( C  i^i  B )  C_  B  /\  ( D  i^i  B ) 
C_  B ) )
4945, 46, 48mpbir2an 911 . . . . . . 7  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
5049a1i 11 . . . . . 6  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B )
515, 6, 213pm3.2i 1166 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )
52 mdsl3 25857 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
) )  ->  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
5351, 52mpan 670 . . . . . 6  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
)  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5441, 44, 50, 53syl3an 1261 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5540, 54sseqtrd 3492 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
56553expb 1189 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5710, 56sylan2b 475 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
583, 4, 6lediri 25077 . . 3  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  (
( C  vH  D
)  i^i  B )
5958a1i 11 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  ( ( C  vH  D )  i^i 
B ) )
6057, 59eqssd 3473 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3427    C_ wss 3428   class class class wbr 4392  (class class class)co 6192   CHcch 24468    vH chj 24472    MH cmd 24505    MH* cdmd 24506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cc 8707  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463  ax-addf 9464  ax-mulf 9465  ax-hilex 24538  ax-hfvadd 24539  ax-hvcom 24540  ax-hvass 24541  ax-hv0cl 24542  ax-hvaddid 24543  ax-hfvmul 24544  ax-hvmulid 24545  ax-hvmulass 24546  ax-hvdistr1 24547  ax-hvdistr2 24548  ax-hvmul0 24549  ax-hfi 24618  ax-his1 24621  ax-his2 24622  ax-his3 24623  ax-his4 24624  ax-hcompl 24741
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-omul 7027  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-fi 7764  df-sup 7794  df-oi 7827  df-card 8212  df-acn 8215  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-q 11057  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-ioo 11407  df-ico 11409  df-icc 11410  df-fz 11541  df-fzo 11652  df-fl 11745  df-seq 11910  df-exp 11969  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-rlim 13071  df-sum 13268  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-starv 14357  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-hom 14366  df-cco 14367  df-rest 14465  df-topn 14466  df-0g 14484  df-gsum 14485  df-topgen 14486  df-pt 14487  df-prds 14490  df-xrs 14544  df-qtop 14549  df-imas 14550  df-xps 14552  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-submnd 15569  df-mulg 15652  df-cntz 15939  df-cmn 16385  df-psmet 17920  df-xmet 17921  df-met 17922  df-bl 17923  df-mopn 17924  df-fbas 17925  df-fg 17926  df-cnfld 17930  df-top 18621  df-bases 18623  df-topon 18624  df-topsp 18625  df-cld 18741  df-ntr 18742  df-cls 18743  df-nei 18820  df-cn 18949  df-cnp 18950  df-lm 18951  df-haus 19037  df-tx 19253  df-hmeo 19446  df-fil 19537  df-fm 19629  df-flim 19630  df-flf 19631  df-xms 20013  df-ms 20014  df-tms 20015  df-cfil 20884  df-cau 20885  df-cmet 20886  df-grpo 23815  df-gid 23816  df-ginv 23817  df-gdiv 23818  df-ablo 23906  df-subgo 23926  df-vc 24061  df-nv 24107  df-va 24110  df-ba 24111  df-sm 24112  df-0v 24113  df-vs 24114  df-nmcv 24115  df-ims 24116  df-dip 24233  df-ssp 24257  df-ph 24350  df-cbn 24401  df-hnorm 24507  df-hba 24508  df-hvsub 24510  df-hlim 24511  df-hcau 24512  df-sh 24746  df-ch 24761  df-oc 24792  df-ch0 24793  df-shs 24848  df-chj 24850  df-md 25821  df-dmd 25822
This theorem is referenced by:  mdslmd1lem1  25866  mdslmd1lem2  25867
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