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Theorem mdbr 22874
Description: Binary relation expressing  <. A ,  B >. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdbr  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem mdbr
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 685 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 oveq2 5866 . . . . . . . 8  |-  ( y  =  A  ->  (
x  vH  y )  =  ( x  vH  A ) )
43ineq1d 3369 . . . . . . 7  |-  ( y  =  A  ->  (
( x  vH  y
)  i^i  z )  =  ( ( x  vH  A )  i^i  z ) )
5 ineq1 3363 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
65oveq2d 5874 . . . . . . 7  |-  ( y  =  A  ->  (
x  vH  ( y  i^i  z ) )  =  ( x  vH  ( A  i^i  z ) ) )
74, 6eqeq12d 2297 . . . . . 6  |-  ( y  =  A  ->  (
( ( x  vH  y )  i^i  z
)  =  ( x  vH  ( y  i^i  z ) )  <->  ( (
x  vH  A )  i^i  z )  =  ( x  vH  ( A  i^i  z ) ) ) )
87imbi2d 307 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  z  ->  ( ( x  vH  y )  i^i  z
)  =  ( x  vH  ( y  i^i  z ) ) )  <-> 
( x  C_  z  ->  ( ( x  vH  A )  i^i  z
)  =  ( x  vH  ( A  i^i  z ) ) ) ) )
98ralbidv 2563 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  CH  (
x  C_  z  ->  ( ( x  vH  y
)  i^i  z )  =  ( x  vH  ( y  i^i  z
) ) )  <->  A. x  e.  CH  ( x  C_  z  ->  ( ( x  vH  A )  i^i  z )  =  ( x  vH  ( A  i^i  z ) ) ) ) )
102, 9anbi12d 691 . . 3  |-  ( y  =  A  ->  (
( ( y  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( x  C_  z  ->  ( ( x  vH  y )  i^i  z )  =  ( x  vH  ( y  i^i  z ) ) ) )  <->  ( ( A  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( x 
C_  z  ->  (
( x  vH  A
)  i^i  z )  =  ( x  vH  ( A  i^i  z
) ) ) ) ) )
11 eleq1 2343 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 684 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq2 3200 . . . . . 6  |-  ( z  =  B  ->  (
x  C_  z  <->  x  C_  B
) )
14 ineq2 3364 . . . . . . 7  |-  ( z  =  B  ->  (
( x  vH  A
)  i^i  z )  =  ( ( x  vH  A )  i^i 
B ) )
15 ineq2 3364 . . . . . . . 8  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1615oveq2d 5874 . . . . . . 7  |-  ( z  =  B  ->  (
x  vH  ( A  i^i  z ) )  =  ( x  vH  ( A  i^i  B ) ) )
1714, 16eqeq12d 2297 . . . . . 6  |-  ( z  =  B  ->  (
( ( x  vH  A )  i^i  z
)  =  ( x  vH  ( A  i^i  z ) )  <->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )
1813, 17imbi12d 311 . . . . 5  |-  ( z  =  B  ->  (
( x  C_  z  ->  ( ( x  vH  A )  i^i  z
)  =  ( x  vH  ( A  i^i  z ) ) )  <-> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
1918ralbidv 2563 . . . 4  |-  ( z  =  B  ->  ( A. x  e.  CH  (
x  C_  z  ->  ( ( x  vH  A
)  i^i  z )  =  ( x  vH  ( A  i^i  z
) ) )  <->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  A )  i^i 
B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
2012, 19anbi12d 691 . . 3  |-  ( z  =  B  ->  (
( ( A  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( x  C_  z  ->  ( ( x  vH  A )  i^i  z )  =  ( x  vH  ( A  i^i  z ) ) ) )  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) ) )
21 df-md 22860 . . 3  |-  MH  =  { <. y ,  z
>.  |  ( (
y  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( x  C_  z  -> 
( ( x  vH  y )  i^i  z
)  =  ( x  vH  ( y  i^i  z ) ) ) ) }
2210, 20, 21brabg 4284 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( x  C_  B  -> 
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) ) )
2322bianabs 850 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513    MH cmd 21546
This theorem is referenced by:  mdi  22875  mdbr2  22876  mdbr3  22877  dmdmd  22880  mddmd2  22889  mdsl1i  22901
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
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-md 22860
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