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Theorem mdbr 22890
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 2356 . . . . 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 5882 . . . . . . . 8  |-  ( y  =  A  ->  (
x  vH  y )  =  ( x  vH  A ) )
43ineq1d 3382 . . . . . . 7  |-  ( y  =  A  ->  (
( x  vH  y
)  i^i  z )  =  ( ( x  vH  A )  i^i  z ) )
5 ineq1 3376 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
65oveq2d 5890 . . . . . . 7  |-  ( y  =  A  ->  (
x  vH  ( y  i^i  z ) )  =  ( x  vH  ( A  i^i  z ) ) )
74, 6eqeq12d 2310 . . . . . 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 2576 . . . 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 2356 . . . . 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 3213 . . . . . 6  |-  ( z  =  B  ->  (
x  C_  z  <->  x  C_  B
) )
14 ineq2 3377 . . . . . . 7  |-  ( z  =  B  ->  (
( x  vH  A
)  i^i  z )  =  ( ( x  vH  A )  i^i 
B ) )
15 ineq2 3377 . . . . . . . 8  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1615oveq2d 5890 . . . . . . 7  |-  ( z  =  B  ->  (
x  vH  ( A  i^i  z ) )  =  ( x  vH  ( A  i^i  B ) ) )
1714, 16eqeq12d 2310 . . . . . 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 2576 . . . 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 22876 . . 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 4300 . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   CHcch 21525    vH chj 21529    MH cmd 21562
This theorem is referenced by:  mdi  22891  mdbr2  22892  mdbr3  22893  dmdmd  22896  mddmd2  22905  mdsl1i  22917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279  df-ov 5877  df-md 22876
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