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Theorem mdsymlem6 23004
Description: Lemma for mdsymi 23007. This is the converse direction of Lemma 4(i) of [Maeda] p. 168, and is based on the proof of Theorem 1(d) to (e) of [Maeda] p. 167. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdsymlem1.1  |-  A  e. 
CH
mdsymlem1.2  |-  B  e. 
CH
mdsymlem1.3  |-  C  =  ( A  vH  p
)
Assertion
Ref Expression
mdsymlem6  |-  ( A. p  e. HAtoms  ( p  C_  ( A  vH  B
)  ->  E. q  e. HAtoms  E. r  e. HAtoms  (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) ) )  ->  B  MH*  A )
Distinct variable groups:    r, q, C    q, p, r, A    B, p, q, r
Allowed substitution hint:    C( p)

Proof of Theorem mdsymlem6
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 mdsymlem1.1 . . . . . . . . . . . . 13  |-  A  e. 
CH
2 mdsymlem1.2 . . . . . . . . . . . . 13  |-  B  e. 
CH
31, 2chjcomi 22063 . . . . . . . . . . . 12  |-  ( A  vH  B )  =  ( B  vH  A
)
43sseq2i 3216 . . . . . . . . . . 11  |-  ( p 
C_  ( A  vH  B )  <->  p  C_  ( B  vH  A ) )
54anbi2i 675 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( A  vH  B ) )  <->  ( p  C_  c  /\  p  C_  ( B  vH  A ) ) )
6 ssin 3404 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( B  vH  A ) )  <->  p  C_  (
c  i^i  ( B  vH  A ) ) )
75, 6bitri 240 . . . . . . . . 9  |-  ( ( p  C_  c  /\  p  C_  ( A  vH  B ) )  <->  p  C_  (
c  i^i  ( B  vH  A ) ) )
8 mdsymlem1.3 . . . . . . . . . . . . . . . 16  |-  C  =  ( A  vH  p
)
91, 2, 8mdsymlem5 23003 . . . . . . . . . . . . . . 15  |-  ( ( q  e. HAtoms  /\  r  e. HAtoms )  ->  ( -.  q  =  p  ->  ( ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) )  ->  ( (
( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( p  C_  c  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) ) ) ) )
10 sseq1 3212 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  p  ->  (
q  C_  A  <->  p  C_  A
) )
11 chincl 22094 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( c  e.  CH  /\  B  e.  CH )  ->  ( c  i^i  B
)  e.  CH )
122, 11mpan2 652 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c  e.  CH  ->  (
c  i^i  B )  e.  CH )
13 chub2 22103 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  CH  /\  ( c  i^i  B
)  e.  CH )  ->  A  C_  ( (
c  i^i  B )  vH  A ) )
141, 12, 13sylancr 644 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  e.  CH  ->  A  C_  ( ( c  i^i 
B )  vH  A
) )
15 sstr2 3199 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p 
C_  A  ->  ( A  C_  ( ( c  i^i  B )  vH  A )  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) )
1614, 15syl5 28 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p 
C_  A  ->  (
c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) )
1710, 16syl6bi 219 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  (
q  C_  A  ->  ( c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) ) )
1817imp3a 420 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
( q  C_  A  /\  c  e.  CH )  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) )
1918a1i 10 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p 
C_  c  ->  (
q  =  p  -> 
( ( q  C_  A  /\  c  e.  CH )  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2019com13 74 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  C_  A  /\  c  e.  CH )  ->  ( q  =  p  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2120adantrr 697 . . . . . . . . . . . . . . . . . . 19  |-  ( ( q  C_  A  /\  ( c  e.  CH  /\  A  C_  c )
)  ->  ( q  =  p  ->  ( p 
C_  c  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) ) )
2221ad2ant2r 727 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( q  C_  A  /\  r  C_  B )  /\  ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms ) )  ->  (
q  =  p  -> 
( p  C_  c  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) ) )
2322adantll 694 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) )  /\  ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms ) )  ->  ( q  =  p  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2423com12 27 . . . . . . . . . . . . . . . 16  |-  ( q  =  p  ->  (
( ( p  C_  ( q  vH  r
)  /\  ( q  C_  A  /\  r  C_  B ) )  /\  ( ( c  e. 
CH  /\  A  C_  c
)  /\  p  e. HAtoms ) )  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2524exp3a 425 . . . . . . . . . . . . . . 15  |-  ( q  =  p  ->  (
( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) )  ->  ( (
( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( p  C_  c  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) ) ) )
269, 25pm2.61d2 152 . . . . . . . . . . . . . 14  |-  ( ( q  e. HAtoms  /\  r  e. HAtoms )  ->  ( (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) )  ->  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  ->  ( p  C_  c  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) ) ) )
2726rexlimivv 2685 . . . . . . . . . . . . 13  |-  ( E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) )  ->  ( (
( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( p  C_  c  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) ) )
2827com12 27 . . . . . . . . . . . 12  |-  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( E. q  e. HAtoms  E. r  e. HAtoms  ( p 
C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) )  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2928imim2d 48 . . . . . . . . . . 11  |-  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  (
p  C_  ( A  vH  B )  ->  (
p  C_  c  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) ) ) )
3029com34 77 . . . . . . . . . 10  |-  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  (
p  C_  c  ->  ( p  C_  ( A  vH  B )  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) ) ) )
3130imp4b 573 . . . . . . . . 9  |-  ( ( ( ( c  e. 
CH  /\  A  C_  c
)  /\  p  e. HAtoms )  /\  ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) ) )  -> 
( ( p  C_  c  /\  p  C_  ( A  vH  B ) )  ->  p  C_  (
( c  i^i  B
)  vH  A )
) )
327, 31syl5bir 209 . . . . . . . 8  |-  ( ( ( ( c  e. 
CH  /\  A  C_  c
)  /\  p  e. HAtoms )  /\  ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) ) )  -> 
( p  C_  (
c  i^i  ( B  vH  A ) )  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) )
3332ex 423 . . . . . . 7  |-  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  -> 
( ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  (
p  C_  ( c  i^i  ( B  vH  A
) )  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) ) )
3433ralimdva 2634 . . . . . 6  |-  ( ( c  e.  CH  /\  A  C_  c )  -> 
( A. p  e. HAtoms  ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  A. p  e. HAtoms  ( p  C_  (
c  i^i  ( B  vH  A ) )  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) ) )
352, 1chjcli 22052 . . . . . . . . 9  |-  ( B  vH  A )  e. 
CH
36 chincl 22094 . . . . . . . . 9  |-  ( ( c  e.  CH  /\  ( B  vH  A )  e.  CH )  -> 
( c  i^i  ( B  vH  A ) )  e.  CH )
3735, 36mpan2 652 . . . . . . . 8  |-  ( c  e.  CH  ->  (
c  i^i  ( B  vH  A ) )  e. 
CH )
38 chjcl 21952 . . . . . . . . 9  |-  ( ( ( c  i^i  B
)  e.  CH  /\  A  e.  CH )  ->  ( ( c  i^i 
B )  vH  A
)  e.  CH )
3912, 1, 38sylancl 643 . . . . . . . 8  |-  ( c  e.  CH  ->  (
( c  i^i  B
)  vH  A )  e.  CH )
40 chrelat3 22967 . . . . . . . 8  |-  ( ( ( c  i^i  ( B  vH  A ) )  e.  CH  /\  (
( c  i^i  B
)  vH  A )  e.  CH )  ->  (
( c  i^i  ( B  vH  A ) ) 
C_  ( ( c  i^i  B )  vH  A )  <->  A. p  e. HAtoms  ( p  C_  (
c  i^i  ( B  vH  A ) )  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) ) )
4137, 39, 40syl2anc 642 . . . . . . 7  |-  ( c  e.  CH  ->  (
( c  i^i  ( B  vH  A ) ) 
C_  ( ( c  i^i  B )  vH  A )  <->  A. p  e. HAtoms  ( p  C_  (
c  i^i  ( B  vH  A ) )  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) ) )
4241adantr 451 . . . . . 6  |-  ( ( c  e.  CH  /\  A  C_  c )  -> 
( ( c  i^i  ( B  vH  A
) )  C_  (
( c  i^i  B
)  vH  A )  <->  A. p  e. HAtoms  ( p  C_  ( c  i^i  ( B  vH  A ) )  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
4334, 42sylibrd 225 . . . . 5  |-  ( ( c  e.  CH  /\  A  C_  c )  -> 
( A. p  e. HAtoms  ( p  C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  (
c  i^i  ( B  vH  A ) )  C_  ( ( c  i^i 
B )  vH  A
) ) )
4443ex 423 . . . 4  |-  ( c  e.  CH  ->  ( A  C_  c  ->  ( A. p  e. HAtoms  ( p 
C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) ) )  ->  ( c  i^i  ( B  vH  A
) )  C_  (
( c  i^i  B
)  vH  A )
) ) )
4544com3r 73 . . 3  |-  ( A. p  e. HAtoms  ( p  C_  ( A  vH  B
)  ->  E. q  e. HAtoms  E. r  e. HAtoms  (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) ) )  ->  ( c  e.  CH  ->  ( A  C_  c  ->  ( c  i^i  ( B  vH  A
) )  C_  (
( c  i^i  B
)  vH  A )
) ) )
4645ralrimiv 2638 . 2  |-  ( A. p  e. HAtoms  ( p  C_  ( A  vH  B
)  ->  E. q  e. HAtoms  E. r  e. HAtoms  (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) ) )  ->  A. c  e.  CH  ( A  C_  c  ->  ( c  i^i  ( B  vH  A
) )  C_  (
( c  i^i  B
)  vH  A )
) )
47 dmdbr2 22899 . . 3  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  MH*  A  <->  A. c  e.  CH  ( A  C_  c  ->  (
c  i^i  ( B  vH  A ) )  C_  ( ( c  i^i 
B )  vH  A
) ) ) )
482, 1, 47mp2an 653 . 2  |-  ( B 
MH*  A  <->  A. c  e.  CH  ( A  C_  c  -> 
( c  i^i  ( B  vH  A ) ) 
C_  ( ( c  i^i  B )  vH  A ) ) )
4946, 48sylibr 203 1  |-  ( A. p  e. HAtoms  ( p  C_  ( A  vH  B
)  ->  E. q  e. HAtoms  E. r  e. HAtoms  (
p  C_  ( q  vH  r )  /\  (
q  C_  A  /\  r  C_  B ) ) )  ->  B  MH*  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   CHcch 21525    vH chj 21529  HAtomscat 21561    MH* cdmd 21563
This theorem is referenced by:  mdsymlem7  23005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680  ax-hcompl 21797
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cn 16973  df-cnp 16974  df-lm 16975  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cfil 18697  df-cau 18698  df-cmet 18699  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-dip 21290  df-ssp 21314  df-ph 21407  df-cbn 21458  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-hcau 21569  df-sh 21802  df-ch 21817  df-oc 21847  df-ch0 21848  df-shs 21903  df-span 21904  df-chj 21905  df-chsup 21906  df-pjh 21990  df-cv 22875  df-dmd 22877  df-at 22934
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