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Theorem dmdsl3 22911
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 22898 . . . . . 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 588 . . . . 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 1155 . . . 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 422 . . 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 1116 . 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 22101 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
76ineq2d 3383 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  i^i  ( A  vH  B ) )  =  ( C  i^i  ( B  vH  A ) ) )
873adant3 975 . . . 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 3179 . . . . 5  |-  ( C 
C_  ( A  vH  B )  <->  ( C  i^i  ( A  vH  B
) )  =  C )
109biimpi 186 . . . 4  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  ( A  vH  B ) )  =  C )
118, 10sylan9req 2349 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  C  C_  ( A  vH  B ) )  ->  ( C  i^i  ( B  vH  A ) )  =  C )
12113ad2antr3 1122 . 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 2328 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 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   CHcch 21525    vH chj 21529    MH* cdmd 21563
This theorem is referenced by:  mdslle1i  22913  mdslj1i  22915  mdslj2i  22916  mdslmd1lem1  22921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-sh 21802  df-ch 21817  df-chj 21905  df-dmd 22877
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