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Theorem cmbr 26628
Description: Binary relation expressing  A commutes with  B. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cmbr  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )

Proof of Theorem cmbr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2529 . . . . 5  |-  ( x  =  A  ->  (
x  e.  CH  <->  A  e.  CH ) )
21anbi1d 704 . . . 4  |-  ( x  =  A  ->  (
( x  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  y  e.  CH )
) )
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 ineq1 3689 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
5 ineq1 3689 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  ( _|_ `  y ) )  =  ( A  i^i  ( _|_ `  y ) ) )
64, 5oveq12d 6314 . . . . 5  |-  ( x  =  A  ->  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) )
73, 6eqeq12d 2479 . . . 4  |-  ( x  =  A  ->  (
x  =  ( ( x  i^i  y )  vH  ( x  i^i  ( _|_ `  y
) ) )  <->  A  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) ) )
82, 7anbi12d 710 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
CH  /\  y  e.  CH )  /\  x  =  ( ( x  i^i  y )  vH  (
x  i^i  ( _|_ `  y ) ) ) )  <->  ( ( A  e.  CH  /\  y  e.  CH )  /\  A  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y ) ) ) ) ) )
9 eleq1 2529 . . . . 5  |-  ( y  =  B  ->  (
y  e.  CH  <->  B  e.  CH ) )
109anbi2d 703 . . . 4  |-  ( y  =  B  ->  (
( A  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
11 ineq2 3690 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
12 fveq2 5872 . . . . . . 7  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1312ineq2d 3696 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  ( _|_ `  y
) )  =  ( A  i^i  ( _|_ `  B ) ) )
1411, 13oveq12d 6314 . . . . 5  |-  ( y  =  B  ->  (
( A  i^i  y
)  vH  ( A  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
1514eqeq2d 2471 . . . 4  |-  ( y  =  B  ->  ( A  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y ) ) )  <->  A  =  (
( A  i^i  B
)  vH  ( A  i^i  ( _|_ `  B
) ) ) ) )
1610, 15anbi12d 710 . . 3  |-  ( y  =  B  ->  (
( ( A  e. 
CH  /\  y  e.  CH )  /\  A  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) )  <-> 
( ( A  e. 
CH  /\  B  e.  CH )  /\  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) ) ) )
17 df-cm 26627 . . 3  |-  C_H  =  { <. x ,  y
>.  |  ( (
x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
188, 16, 17brabg 4775 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A  =  (
( A  i^i  B
)  vH  ( A  i^i  ( _|_ `  B
) ) ) ) ) )
1918bianabs 880 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CHcch 25972   _|_cort 25973    vH chj 25976    C_H ccm 25979
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-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-cm 26627
This theorem is referenced by:  cmbri  26634  cm2j  26664
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