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Theorem mdsl0 22882
Description: A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdsl0  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( (
( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B )  ->  C  MH  D ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem mdsl0
StepHypRef Expression
1 sstr2 3187 . . . . . . . 8  |-  ( x 
C_  D  ->  ( D  C_  B  ->  x  C_  B ) )
21com12 29 . . . . . . 7  |-  ( D 
C_  B  ->  (
x  C_  D  ->  x 
C_  B ) )
32ad2antlr 709 . . . . . 6  |-  ( ( ( C  C_  A  /\  D  C_  B )  /\  ( A  i^i  B )  =  0H )  ->  ( x  C_  D  ->  x  C_  B
) )
43ad2antlr 709 . . . . 5  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
x  C_  D  ->  x 
C_  B ) )
5 chlej2 22082 . . . . . . . . . . . . . 14  |-  ( ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  C  C_  A )  ->  ( x  vH  C )  C_  (
x  vH  A )
)
6 ss2in 3397 . . . . . . . . . . . . . . 15  |-  ( ( ( x  vH  C
)  C_  ( x  vH  A )  /\  D  C_  B )  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) )
76ex 425 . . . . . . . . . . . . . 14  |-  ( ( x  vH  C ) 
C_  ( x  vH  A )  ->  ( D  C_  B  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) ) )
85, 7syl 17 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  C  C_  A )  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i 
D )  C_  (
( x  vH  A
)  i^i  B )
) )
98ex 425 . . . . . . . . . . . 12  |-  ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  ( C  C_  A  ->  ( D  C_  B  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) ) ) )
1093expia 1155 . . . . . . . . . . 11  |-  ( ( C  e.  CH  /\  A  e.  CH )  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i  D
)  C_  ( (
x  vH  A )  i^i  B ) ) ) ) )
1110ancoms 441 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i  D
)  C_  ( (
x  vH  A )  i^i  B ) ) ) ) )
1211ad2ant2r 729 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( (
x  vH  C )  i^i  D )  C_  (
( x  vH  A
)  i^i  B )
) ) ) )
1312imp43 580 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( C  C_  A  /\  D  C_  B ) )  ->  ( (
x  vH  C )  i^i  D )  C_  (
( x  vH  A
)  i^i  B )
)
1413adantrr 699 . . . . . . 7  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) )
15 oveq2 5827 . . . . . . . . . . . . 13  |-  ( ( A  i^i  B )  =  0H  ->  (
x  vH  ( A  i^i  B ) )  =  ( x  vH  0H ) )
16 chj0 22068 . . . . . . . . . . . . 13  |-  ( x  e.  CH  ->  (
x  vH  0H )  =  x )
1715, 16sylan9eqr 2338 . . . . . . . . . . . 12  |-  ( ( x  e.  CH  /\  ( A  i^i  B )  =  0H )  -> 
( x  vH  ( A  i^i  B ) )  =  x )
1817adantl 454 . . . . . . . . . . 11  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  ( x  vH  ( A  i^i  B
) )  =  x )
19 chincl 22070 . . . . . . . . . . . . 13  |-  ( ( C  e.  CH  /\  D  e.  CH )  ->  ( C  i^i  D
)  e.  CH )
20 chub1 22078 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( C  i^i  D )  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D ) ) )
2120ancoms 441 . . . . . . . . . . . . 13  |-  ( ( ( C  i^i  D
)  e.  CH  /\  x  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D
) ) )
2219, 21sylan 459 . . . . . . . . . . . 12  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  x  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D
) ) )
2322adantrr 699 . . . . . . . . . . 11  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  x  C_  (
x  vH  ( C  i^i  D ) ) )
2418, 23eqsstrd 3213 . . . . . . . . . 10  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  ( x  vH  ( A  i^i  B
) )  C_  (
x  vH  ( C  i^i  D ) ) )
2524adantll 696 . . . . . . . . 9  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( x  e.  CH  /\  ( A  i^i  B )  =  0H ) )  ->  ( x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D
) ) )
2625anassrs 631 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( A  i^i  B
)  =  0H )  ->  ( x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D
) ) )
2726adantrl 698 . . . . . . 7  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D ) ) )
28 sstr2 3187 . . . . . . . . 9  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( A  i^i  B ) ) ) )
29 sstr2 3187 . . . . . . . . 9  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  ( A  i^i  B ) )  C_  (
x  vH  ( C  i^i  D ) )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) )
3028, 29syl6 31 . . . . . . . 8  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  ( A  i^i  B ) )  C_  (
x  vH  ( C  i^i  D ) )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) )
3130com23 74 . . . . . . 7  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D ) )  ->  ( ( ( x  vH  A )  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
3214, 27, 31sylc 58 . . . . . 6  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) )
3332an32s 781 . . . . 5  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) )
344, 33imim12d 70 . . . 4  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) ) )  -> 
( x  C_  D  ->  ( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) )
3534ralimdva 2622 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) )  ->  A. x  e.  CH  ( x  C_  D  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
36 mdbr2 22868 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) ) ) )
3736ad2antrr 708 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  A )  i^i 
B )  C_  (
x  vH  ( A  i^i  B ) ) ) ) )
38 mdbr2 22868 . . . 4  |-  ( ( C  e.  CH  /\  D  e.  CH )  ->  ( C  MH  D  <->  A. x  e.  CH  (
x  C_  D  ->  ( ( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) ) )
3938ad2antlr 709 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( C  MH  D  <->  A. x  e.  CH  ( x  C_  D  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
4035, 37, 393imtr4d 261 . 2  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  MH  B  ->  C  MH  D ) )
4140expimpd 588 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( (
( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B )  ->  C  MH  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2544    i^i cin 3152    C_ wss 3153   class class class wbr 4024  (class class class)co 5819   CHcch 21501    vH chj 21505   0Hc0h 21507    MH cmd 21538
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cc 8056  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvmulass 21579  ax-hvdistr1 21580  ax-hvdistr2 21581  ax-hvmul0 21582  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655  ax-his4 21656  ax-hcompl 21773
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-acn 7570  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-rlim 11957  df-sum 12153  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-cn 16951  df-cnp 16952  df-lm 16953  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cfil 18675  df-cau 18676  df-cmet 18677  df-grpo 20850  df-gid 20851  df-ginv 20852  df-gdiv 20853  df-ablo 20941  df-subgo 20961  df-vc 21094  df-nv 21140  df-va 21143  df-ba 21144  df-sm 21145  df-0v 21146  df-vs 21147  df-nmcv 21148  df-ims 21149  df-dip 21266  df-ssp 21290  df-ph 21383  df-cbn 21434  df-hnorm 21540  df-hba 21541  df-hvsub 21543  df-hlim 21544  df-hcau 21545  df-sh 21778  df-ch 21793  df-oc 21823  df-ch0 21824  df-shs 21879  df-chj 21881  df-md 22852
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