HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  dmdsl3 Unicode version

Theorem dmdsl3 22887
Description: Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdsl3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )

Proof of Theorem dmdsl3
StepHypRef Expression
1 dmdi 22874 . . . . . 6  |-  ( ( ( B  e.  CH  /\  A  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C ) )  ->  ( ( C  i^i  B )  vH  A )  =  ( C  i^i  ( B  vH  A ) ) )
21exp32 591 . . . . 5  |-  ( ( B  e.  CH  /\  A  e.  CH  /\  C  e.  CH )  ->  ( B  MH*  A  ->  ( A  C_  C  ->  (
( C  i^i  B
)  vH  A )  =  ( C  i^i  ( B  vH  A ) ) ) ) )
323com12 1160 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  MH*  A  ->  ( A  C_  C  ->  (
( C  i^i  B
)  vH  A )  =  ( C  i^i  ( B  vH  A ) ) ) ) )
43imp32 424 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C ) )  ->  ( ( C  i^i  B )  vH  A )  =  ( C  i^i  ( B  vH  A ) ) )
543adantr3 1121 . 2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  ( C  i^i  ( B  vH  A ) ) )
6 chjcom 22077 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
76ineq2d 3371 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  i^i  ( A  vH  B ) )  =  ( C  i^i  ( B  vH  A ) ) )
873adant3 980 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( C  i^i  ( A  vH  B ) )  =  ( C  i^i  ( B  vH  A ) ) )
9 df-ss 3167 . . . . 5  |-  ( C 
C_  ( A  vH  B )  <->  ( C  i^i  ( A  vH  B
) )  =  C )
109biimpi 188 . . . 4  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  ( A  vH  B ) )  =  C )
118, 10sylan9req 2337 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  C  C_  ( A  vH  B ) )  ->  ( C  i^i  ( B  vH  A ) )  =  C )
12113ad2antr3 1127 . 2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( C  i^i  ( B  vH  A ) )  =  C )
135, 12eqtrd 2316 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    i^i cin 3152    C_ wss 3153   class class class wbr 4024  (class class class)co 5819   CHcch 21501    vH chj 21505    MH* cdmd 21539
This theorem is referenced by:  mdslle1i  22889  mdslj1i  22891  mdslj2i  22892  mdslmd1lem1  22897
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-hilex 21571
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  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-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-sh 21778  df-ch 21793  df-chj 21881  df-dmd 22853
  Copyright terms: Public domain W3C validator