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

Theorem dmdsl3 22895
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 22882 . . . . . 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 22085 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
76ineq2d 3370 . . . . 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 3166 . . . . 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 2336 . . 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 2315 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513    MH* cdmd 21547
This theorem is referenced by:  mdslle1i  22897  mdslj1i  22899  mdslj2i  22900  mdslmd1lem1  22905
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sh 21786  df-ch 21801  df-chj 21889  df-dmd 22861
  Copyright terms: Public domain W3C validator