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Theorem mdsymlem6 22981
Description: Lemma for mdsymi 22984. 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
Dummy variable  c is distinct from all other variables.
Allowed substitution hint:    C( p)

Proof of Theorem mdsymlem6
StepHypRef Expression
1 mdsymlem1.1 . . . . . . . . . . . . 13  |-  A  e. 
CH
2 mdsymlem1.2 . . . . . . . . . . . . 13  |-  B  e. 
CH
31, 2chjcomi 22040 . . . . . . . . . . . 12  |-  ( A  vH  B )  =  ( B  vH  A
)
43sseq2i 3205 . . . . . . . . . . 11  |-  ( p 
C_  ( A  vH  B )  <->  p  C_  ( B  vH  A ) )
54anbi2i 677 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( A  vH  B ) )  <->  ( p  C_  c  /\  p  C_  ( B  vH  A ) ) )
6 ssin 3393 . . . . . . . . . 10  |-  ( ( p  C_  c  /\  p  C_  ( B  vH  A ) )  <->  p  C_  (
c  i^i  ( B  vH  A ) ) )
75, 6bitri 242 . . . . . . . . 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 22980 . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  p  ->  (
q  C_  A  <->  p  C_  A
) )
11 chincl 22071 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( c  e.  CH  /\  B  e.  CH )  ->  ( c  i^i  B
)  e.  CH )
122, 11mpan2 654 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c  e.  CH  ->  (
c  i^i  B )  e.  CH )
13 chub2 22080 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  CH  /\  ( c  i^i  B
)  e.  CH )  ->  A  C_  ( (
c  i^i  B )  vH  A ) )
141, 12, 13sylancr 646 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  e.  CH  ->  A  C_  ( ( c  i^i 
B )  vH  A
) )
15 sstr2 3188 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p 
C_  A  ->  ( A  C_  ( ( c  i^i  B )  vH  A )  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) )
1614, 15syl5 30 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p 
C_  A  ->  (
c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) )
1710, 16syl6bi 221 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  (
q  C_  A  ->  ( c  e.  CH  ->  p 
C_  ( ( c  i^i  B )  vH  A ) ) ) )
1817imp3a 422 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
( q  C_  A  /\  c  e.  CH )  ->  p  C_  ( (
c  i^i  B )  vH  A ) ) )
1918a1i 12 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p 
C_  c  ->  (
q  =  p  -> 
( ( q  C_  A  /\  c  e.  CH )  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2019com13 76 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  C_  A  /\  c  e.  CH )  ->  ( q  =  p  ->  ( p  C_  c  ->  p  C_  (
( c  i^i  B
)  vH  A )
) ) )
2120adantrr 699 . . . . . . . . . . . . . . . . . . 19  |-  ( ( q  C_  A  /\  ( c  e.  CH  /\  A  C_  c )
)  ->  ( q  =  p  ->  ( p 
C_  c  ->  p  C_  ( ( c  i^i 
B )  vH  A
) ) ) )
2221ad2ant2r 729 . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . 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 29 . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . 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 154 . . . . . . . . . . . . . 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 2674 . . . . . . . . . . . . 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 29 . . . . . . . . . . . 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 50 . . . . . . . . . . 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 79 . . . . . . . . . 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 575 . . . . . . . . 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 211 . . . . . . . 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 425 . . . . . . 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 2623 . . . . . 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 22029 . . . . . . . . 9  |-  ( B  vH  A )  e. 
CH
36 chincl 22071 . . . . . . . . 9  |-  ( ( c  e.  CH  /\  ( B  vH  A )  e.  CH )  -> 
( c  i^i  ( B  vH  A ) )  e.  CH )
3735, 36mpan2 654 . . . . . . . 8  |-  ( c  e.  CH  ->  (
c  i^i  ( B  vH  A ) )  e. 
CH )
38 chjcl 21929 . . . . . . . . 9  |-  ( ( ( c  i^i  B
)  e.  CH  /\  A  e.  CH )  ->  ( ( c  i^i 
B )  vH  A
)  e.  CH )
3912, 1, 38sylancl 645 . . . . . . . 8  |-  ( c  e.  CH  ->  (
( c  i^i  B
)  vH  A )  e.  CH )
40 chrelat3 22944 . . . . . . . 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 644 . . . . . . 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 453 . . . . . 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 227 . . . . 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 425 . . . 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 75 . . 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 2627 . 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 22876 . . 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 655 . 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 205 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2545   E.wrex 2546    i^i cin 3153    C_ wss 3154   class class class wbr 4025  (class class class)co 5820   CHcch 21502    vH chj 21506  HAtomscat 21538    MH* cdmd 21540
This theorem is referenced by:  mdsymlem7  22982
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 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cc 8057  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813  ax-hilex 21572  ax-hfvadd 21573  ax-hvcom 21574  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvmulass 21580  ax-hvdistr1 21581  ax-hvdistr2 21582  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657  ax-hcompl 21774
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10655  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-seq 11042  df-exp 11100  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-rlim 11958  df-sum 12154  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-submnd 14411  df-mulg 14487  df-cntz 14788  df-cmn 15086  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-cn 16952  df-cnp 16953  df-lm 16954  df-haus 17038  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cfil 18676  df-cau 18677  df-cmet 18678  df-grpo 20851  df-gid 20852  df-ginv 20853  df-gdiv 20854  df-ablo 20942  df-subgo 20962  df-vc 21095  df-nv 21141  df-va 21144  df-ba 21145  df-sm 21146  df-0v 21147  df-vs 21148  df-nmcv 21149  df-ims 21150  df-dip 21267  df-ssp 21291  df-ph 21384  df-cbn 21435  df-hnorm 21541  df-hba 21542  df-hvsub 21544  df-hlim 21545  df-hcau 21546  df-sh 21779  df-ch 21794  df-oc 21824  df-ch0 21825  df-shs 21880  df-span 21881  df-chj 21882  df-chsup 21883  df-pjh 21967  df-cv 22852  df-dmd 22854  df-at 22911
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