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Theorem mdbr 22835
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
StepHypRef Expression
1 eleq1 2318 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 688 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 oveq2 5800 . . . . . . . 8  |-  ( y  =  A  ->  (
x  vH  y )  =  ( x  vH  A ) )
43ineq1d 3344 . . . . . . 7  |-  ( y  =  A  ->  (
( x  vH  y
)  i^i  z )  =  ( ( x  vH  A )  i^i  z ) )
5 ineq1 3338 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
65oveq2d 5808 . . . . . . 7  |-  ( y  =  A  ->  (
x  vH  ( y  i^i  z ) )  =  ( x  vH  ( A  i^i  z ) ) )
74, 6eqeq12d 2272 . . . . . 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 309 . . . . 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 2538 . . . 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 694 . . 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 2318 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 687 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq2 3175 . . . . . 6  |-  ( z  =  B  ->  (
x  C_  z  <->  x  C_  B
) )
14 ineq2 3339 . . . . . . 7  |-  ( z  =  B  ->  (
( x  vH  A
)  i^i  z )  =  ( ( x  vH  A )  i^i 
B ) )
15 ineq2 3339 . . . . . . . 8  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1615oveq2d 5808 . . . . . . 7  |-  ( z  =  B  ->  (
x  vH  ( A  i^i  z ) )  =  ( x  vH  ( A  i^i  B ) ) )
1714, 16eqeq12d 2272 . . . . . 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 313 . . . . 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 2538 . . . 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 694 . . 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 22821 . . 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 4256 . 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 855 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    i^i cin 3126    C_ wss 3127   class class class wbr 3997  (class class class)co 5792   CHcch 21470    vH chj 21474    MH cmd 21507
This theorem is referenced by:  mdi  22836  mdbr2  22837  mdbr3  22838  dmdmd  22841  mddmd2  22850  mdsl1i  22862
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795  df-md 22821
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