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Theorem dmdsl3 26938
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 26925 . . . . . 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 605 . . . . 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 1200 . . . 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 433 . . 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 1157 . 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 26128 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
76ineq2d 3700 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  i^i  ( A  vH  B ) )  =  ( C  i^i  ( B  vH  A ) ) )
873adant3 1016 . . . 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 3490 . . . . 5  |-  ( C 
C_  ( A  vH  B )  <->  ( C  i^i  ( A  vH  B
) )  =  C )
109biimpi 194 . . . 4  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  ( A  vH  B ) )  =  C )
118, 10sylan9req 2529 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  C  C_  ( A  vH  B ) )  ->  ( C  i^i  ( B  vH  A ) )  =  C )
12113ad2antr3 1163 . 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 2508 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   class class class wbr 4447  (class class class)co 6284   CHcch 25550    vH chj 25554    MH* cdmd 25588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-sh 25828  df-ch 25843  df-chj 25932  df-dmd 26904
This theorem is referenced by:  mdslle1i  26940  mdslj1i  26942  mdslj2i  26943  mdslmd1lem1  26948
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