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Theorem dmdbr 25701
Description: Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdbr  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem dmdbr
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2501 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 704 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 ineq2 3544 . . . . . . . 8  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
43oveq1d 6104 . . . . . . 7  |-  ( y  =  A  ->  (
( x  i^i  y
)  vH  z )  =  ( ( x  i^i  A )  vH  z ) )
5 oveq1 6096 . . . . . . . 8  |-  ( y  =  A  ->  (
y  vH  z )  =  ( A  vH  z ) )
65ineq2d 3550 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  ( y  vH  z ) )  =  ( x  i^i  ( A  vH  z ) ) )
74, 6eqeq12d 2455 . . . . . 6  |-  ( y  =  A  ->  (
( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) )  <->  ( (
x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) )
87imbi2d 316 . . . . 5  |-  ( y  =  A  ->  (
( z  C_  x  ->  ( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) ) )  <-> 
( z  C_  x  ->  ( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) ) ) ) )
98ralbidv 2733 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  CH  (
z  C_  x  ->  ( ( x  i^i  y
)  vH  z )  =  ( x  i^i  ( y  vH  z
) ) )  <->  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) ) )
102, 9anbi12d 710 . . 3  |-  ( y  =  A  ->  (
( ( y  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  y )  vH  z )  =  ( x  i^i  ( y  vH  z ) ) ) )  <->  ( ( A  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z 
C_  x  ->  (
( x  i^i  A
)  vH  z )  =  ( x  i^i  ( A  vH  z
) ) ) ) ) )
11 eleq1 2501 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 703 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq1 3375 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
14 oveq2 6097 . . . . . . 7  |-  ( z  =  B  ->  (
( x  i^i  A
)  vH  z )  =  ( ( x  i^i  A )  vH  B ) )
15 oveq2 6097 . . . . . . . 8  |-  ( z  =  B  ->  ( A  vH  z )  =  ( A  vH  B
) )
1615ineq2d 3550 . . . . . . 7  |-  ( z  =  B  ->  (
x  i^i  ( A  vH  z ) )  =  ( x  i^i  ( A  vH  B ) ) )
1714, 16eqeq12d 2455 . . . . . 6  |-  ( z  =  B  ->  (
( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) )  <->  ( (
x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) )
1813, 17imbi12d 320 . . . . 5  |-  ( z  =  B  ->  (
( z  C_  x  ->  ( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) ) )  <-> 
( B  C_  x  ->  ( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) ) ) )
1918ralbidv 2733 . . . 4  |-  ( z  =  B  ->  ( A. x  e.  CH  (
z  C_  x  ->  ( ( x  i^i  A
)  vH  z )  =  ( x  i^i  ( A  vH  z
) ) )  <->  A. x  e.  CH  ( B  C_  x  ->  ( ( x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) ) )
2012, 19anbi12d 710 . . 3  |-  ( z  =  B  ->  (
( ( A  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) )  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( B 
C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) ) ) ) )
21 df-dmd 25683 . . 3  |-  MH*  =  { <. y ,  z
>.  |  ( (
y  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) ) ) ) }
2210, 20, 21brabg 4606 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( B  C_  x  -> 
( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) ) ) ) )
2322bianabs 875 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713    i^i cin 3325    C_ wss 3326   class class class wbr 4290  (class class class)co 6089   CHcch 24329    vH chj 24333    MH* cdmd 24367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-iota 5379  df-fv 5424  df-ov 6092  df-dmd 25683
This theorem is referenced by:  dmdmd  25702  dmdi  25704  dmdbr2  25705  dmdbr3  25707  mddmd2  25711
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