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

Theorem dmdsl3 22820
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 22807 . . . . . 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 22010 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
76ineq2d 3312 . . . . 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 3108 . . . . 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 2309 . . 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 2288 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 1619    e. wcel 1621    i^i cin 3093    C_ wss 3094   class class class wbr 3963  (class class class)co 5757   CHcch 21434    vH chj 21438    MH* cdmd 21472
This theorem is referenced by:  mdslle1i  22822  mdslj1i  22824  mdslj2i  22825  mdslmd1lem1  22830
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449  ax-hilex 21504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-sh 21711  df-ch 21726  df-chj 21814  df-dmd 22786
  Copyright terms: Public domain W3C validator