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Theorem mdsymlem6 28060
Description: Lemma for mdsymi 28063. 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 27120 . . . . . . . . . . . 12  |-  ( A  vH  B )  =  ( B  vH  A
)
43sseq2i 3489 . . . . . . . . . . 11  |-  ( p 
C_  ( A  vH  B )  <->  p  C_  ( B  vH  A ) )
54anbi2i 698 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( A  vH  B ) )  <->  ( p  C_  c  /\  p  C_  ( B  vH  A ) ) )
6 ssin 3684 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( B  vH  A ) )  <->  p  C_  (
c  i^i  ( B  vH  A ) ) )
75, 6bitri 252 . . . . . . . . 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 28059 . . . . . . . . . . . . . . 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 3485 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  p  ->  (
q  C_  A  <->  p  C_  A
) )
11 chincl 27151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( c  e.  CH  /\  B  e.  CH )  ->  ( c  i^i  B
)  e.  CH )
122, 11mpan2 675 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c  e.  CH  ->  (
c  i^i  B )  e.  CH )
13 chub2 27160 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  CH  /\  ( c  i^i  B
)  e.  CH )  ->  A  C_  ( (
c  i^i  B )  vH  A ) )
141, 12, 13sylancr 667 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  e.  CH  ->  A  C_  ( ( c  i^i 
B )  vH  A
) )
15 sstr2 3471 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p 
C_  A  ->  ( A  C_  ( ( c  i^i  B )  vH  A )  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) )
1614, 15syl5 33 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p 
C_  A  ->  (
c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) )
1710, 16syl6bi 231 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  (
q  C_  A  ->  ( c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) ) )
1817impd 432 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
( q  C_  A  /\  c  e.  CH )  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) )
1918a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p 
C_  c  ->  (
q  =  p  -> 
( ( q  C_  A  /\  c  e.  CH )  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2019com13 83 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  C_  A  /\  c  e.  CH )  ->  ( q  =  p  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2120adantrr 721 . . . . . . . . . . . . . . . . . . 19  |-  ( ( q  C_  A  /\  ( c  e.  CH  /\  A  C_  c )
)  ->  ( q  =  p  ->  ( p 
C_  c  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) ) )
2221ad2ant2r 751 . . . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . . 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 )
) ) )
2524expd 437 . . . . . . . . . . . . . . 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 163 . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . 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 32 . . . . . . . . . . . 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 54 . . . . . . . . . . 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 86 . . . . . . . . . 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 593 . . . . . . . . 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 221 . . . . . . . 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 435 . . . . . . 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 2830 . . . . . 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 27109 . . . . . . . . 9  |-  ( B  vH  A )  e. 
CH
36 chincl 27151 . . . . . . . . 9  |-  ( ( c  e.  CH  /\  ( B  vH  A )  e.  CH )  -> 
( c  i^i  ( B  vH  A ) )  e.  CH )
3735, 36mpan2 675 . . . . . . . 8  |-  ( c  e.  CH  ->  (
c  i^i  ( B  vH  A ) )  e. 
CH )
38 chjcl 27009 . . . . . . . . 9  |-  ( ( ( c  i^i  B
)  e.  CH  /\  A  e.  CH )  ->  ( ( c  i^i 
B )  vH  A
)  e.  CH )
3912, 1, 38sylancl 666 . . . . . . . 8  |-  ( c  e.  CH  ->  (
( c  i^i  B
)  vH  A )  e.  CH )
40 chrelat3 28023 . . . . . . . 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 665 . . . . . . 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 466 . . . . . 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 237 . . . . 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 435 . . . 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 82 . . 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 2834 . 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 27955 . . 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 676 . 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 215 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772    i^i cin 3435    C_ wss 3436   class class class wbr 4423  (class class class)co 6306   CHcch 26581    vH chj 26585  HAtomscat 26617    MH* cdmd 26619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cc 8873  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627  ax-hilex 26651  ax-hfvadd 26652  ax-hvcom 26653  ax-hvass 26654  ax-hv0cl 26655  ax-hvaddid 26656  ax-hfvmul 26657  ax-hvmulid 26658  ax-hvmulass 26659  ax-hvdistr1 26660  ax-hvdistr2 26661  ax-hvmul0 26662  ax-hfi 26731  ax-his1 26734  ax-his2 26735  ax-his3 26736  ax-his4 26737  ax-hcompl 26854
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-omul 7199  df-er 7375  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-fi 7935  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-acn 8385  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-ioo 11647  df-ico 11649  df-icc 11650  df-fz 11793  df-fzo 11924  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19920  df-bases 19921  df-topon 19922  df-topsp 19923  df-cld 20033  df-ntr 20034  df-cls 20035  df-nei 20113  df-cn 20242  df-cnp 20243  df-lm 20244  df-haus 20330  df-tx 20576  df-hmeo 20769  df-fil 20860  df-fm 20952  df-flim 20953  df-flf 20954  df-xms 21334  df-ms 21335  df-tms 21336  df-cfil 22224  df-cau 22225  df-cmet 22226  df-grpo 25918  df-gid 25919  df-ginv 25920  df-gdiv 25921  df-ablo 26009  df-subgo 26029  df-vc 26164  df-nv 26210  df-va 26213  df-ba 26214  df-sm 26215  df-0v 26216  df-vs 26217  df-nmcv 26218  df-ims 26219  df-dip 26336  df-ssp 26360  df-ph 26453  df-cbn 26504  df-hnorm 26620  df-hba 26621  df-hvsub 26623  df-hlim 26624  df-hcau 26625  df-sh 26859  df-ch 26873  df-oc 26904  df-ch0 26905  df-shs 26960  df-span 26961  df-chj 26962  df-chsup 26963  df-pjh 27047  df-cv 27931  df-dmd 27933  df-at 27990
This theorem is referenced by:  mdsymlem7  28061
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