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Theorem mdbr 27340
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 2529 . . . . 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 oveq2 6304 . . . . . . . 8  |-  ( y  =  A  ->  (
x  vH  y )  =  ( x  vH  A ) )
43ineq1d 3695 . . . . . . 7  |-  ( y  =  A  ->  (
( x  vH  y
)  i^i  z )  =  ( ( x  vH  A )  i^i  z ) )
5 ineq1 3689 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
65oveq2d 6312 . . . . . . 7  |-  ( y  =  A  ->  (
x  vH  ( y  i^i  z ) )  =  ( x  vH  ( A  i^i  z ) ) )
74, 6eqeq12d 2479 . . . . . 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 316 . . . . 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 2896 . . . 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 710 . . 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 2529 . . . . 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 sseq2 3521 . . . . . 6  |-  ( z  =  B  ->  (
x  C_  z  <->  x  C_  B
) )
14 ineq2 3690 . . . . . . 7  |-  ( z  =  B  ->  (
( x  vH  A
)  i^i  z )  =  ( ( x  vH  A )  i^i 
B ) )
15 ineq2 3690 . . . . . . . 8  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1615oveq2d 6312 . . . . . . 7  |-  ( z  =  B  ->  (
x  vH  ( A  i^i  z ) )  =  ( x  vH  ( A  i^i  B ) ) )
1714, 16eqeq12d 2479 . . . . . 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 320 . . . . 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 2896 . . . 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 710 . . 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 27326 . . 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 4775 . 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 880 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   class class class wbr 4456  (class class class)co 6296   CHcch 25973    vH chj 25977    MH cmd 26010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-iota 5557  df-fv 5602  df-ov 6299  df-md 27326
This theorem is referenced by:  mdi  27341  mdbr2  27342  mdbr3  27343  dmdmd  27346  mddmd2  27355  mdsl1i  27367
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