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Theorem mdslj1i 22729
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 3298 . . . . 5  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
21bicomi 195 . . . 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 21866 . . . . . 6  |-  ( A  vH  B )  e. 
CH
83, 4, 7chlubi 21880 . . . . 5  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
98bicomi 195 . . . 4  |-  ( ( C  vH  D ) 
C_  ( A  vH  B )  <->  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )
102, 9anbi12i 681 . . 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 449 . . . . . . . . . 10  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
12 simpl 445 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  C )
13 simpl 445 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  C  C_  ( A  vH  B
) )
145, 6, 33pm3.2i 1135 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
15 dmdsl3 22725 . . . . . . . . . . 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 654 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  A )  =  C )
1711, 12, 13, 16syl3an 1229 . . . . . . . . 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 21869 . . . . . . . . . . 11  |-  ( C  i^i  B )  e. 
CH
194, 6chincli 21869 . . . . . . . . . . 11  |-  ( D  i^i  B )  e. 
CH
2018, 19chub1i 21878 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
2118, 19chjcli 21866 . . . . . . . . . . 11  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH
2218, 21, 5chlej1i 21882 . . . . . . . . . 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 13 . . . . . . . . 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 3134 . . . . . . . 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 449 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
26 simpr 449 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
275, 6, 43pm3.2i 1135 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
28 dmdsl3 22725 . . . . . . . . . . 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 654 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
3011, 25, 26, 29syl3an 1229 . . . . . . . . 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 21879 . . . . . . . . . 10  |-  ( D  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
3219, 21, 5chlej1i 21882 . . . . . . . . . 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 13 . . . . . . . . 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 3134 . . . . . . . 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 520 . . . . . . 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 21866 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  e.  CH
373, 4, 36chlubi 21880 . . . . . . 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 190 . . . . . 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 3301 . . . . . 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 17 . . . . 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 445 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
42 ssrin 3301 . . . . . . . 8  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( C  i^i  B ) )
4342, 20syl6ss 3112 . . . . . . 7  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
4443adantr 453 . . . . . 6  |-  ( ( A  C_  C  /\  A  C_  D )  -> 
( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
45 inss2 3297 . . . . . . . 8  |-  ( C  i^i  B )  C_  B
46 inss2 3297 . . . . . . . 8  |-  ( D  i^i  B )  C_  B
4718, 19, 6chlubi 21880 . . . . . . . . 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 195 . . . . . . . 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 891 . . . . . . 7  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
5049a1i 12 . . . . . 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 1135 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )
52 mdsl3 22726 . . . . . . 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 654 . . . . . 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 1229 . . . . 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 3135 . . . 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 1157 . . 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 463 . 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 21946 . . 3  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  (
( C  vH  D
)  i^i  B )
5958a1i 12 . 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 3117 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 set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    i^i cin 3077    C_ wss 3078   class class class wbr 3920  (class class class)co 5710   CHcch 21339    vH chj 21343    MH cmd 21376    MH* cdmd 21377
This theorem is referenced by:  mdslmd1lem1  22735  mdslmd1lem2  22736
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cc 7945  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494  ax-hcompl 21611
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-acn 7459  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-rlim 11840  df-sum 12036  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-mulg 14327  df-cntz 14628  df-cmn 14926  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-cn 16789  df-cnp 16790  df-lm 16791  df-haus 16875  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cfil 18513  df-cau 18514  df-cmet 18515  df-grpo 20688  df-gid 20689  df-ginv 20690  df-gdiv 20691  df-ablo 20779  df-subgo 20799  df-vc 20932  df-nv 20978  df-va 20981  df-ba 20982  df-sm 20983  df-0v 20984  df-vs 20985  df-nmcv 20986  df-ims 20987  df-dip 21104  df-ssp 21128  df-ph 21221  df-cbn 21272  df-hnorm 21378  df-hba 21379  df-hvsub 21381  df-hlim 21382  df-hcau 21383  df-sh 21616  df-ch 21631  df-oc 21661  df-ch0 21662  df-shs 21717  df-chj 21719  df-md 22690  df-dmd 22691
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