HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdsymlem6 Unicode version

Theorem mdsymlem6 22988
Description: Lemma for mdsymi 22991. 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 22047 . . . . . . . . . . . 12  |-  ( A  vH  B )  =  ( B  vH  A
)
43sseq2i 3203 . . . . . . . . . . 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 3391 . . . . . . . . . 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 22987 . . . . . . . . . . . . . . 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 3199 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( q  =  p  ->  (
q  C_  A  <->  p  C_  A
) )
11 chincl 22078 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22087 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . 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 2621 . . . . . 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 22036 . . . . . . . . 9  |-  ( B  vH  A )  e. 
CH
36 chincl 22078 . . . . . . . . 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 21936 . . . . . . . . 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 22951 . . . . . . . 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 2625 . 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 22883 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513  HAtomscat 21545    MH* cdmd 21547
This theorem is referenced by:  mdsymlem7  22989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-span 21888  df-chj 21889  df-chsup 21890  df-pjh 21974  df-cv 22859  df-dmd 22861  df-at 22918
  Copyright terms: Public domain W3C validator