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Theorem cmbr 23069
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 2490 . . . . 5  |-  ( x  =  A  ->  (
x  e.  CH  <->  A  e.  CH ) )
21anbi1d 686 . . . 4  |-  ( x  =  A  ->  (
( x  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  y  e.  CH )
) )
3 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 ineq1 3522 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
5 ineq1 3522 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  ( _|_ `  y ) )  =  ( A  i^i  ( _|_ `  y ) ) )
64, 5oveq12d 6085 . . . . 5  |-  ( x  =  A  ->  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) )
73, 6eqeq12d 2444 . . . 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 692 . . 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 2490 . . . . 5  |-  ( y  =  B  ->  (
y  e.  CH  <->  B  e.  CH ) )
109anbi2d 685 . . . 4  |-  ( y  =  B  ->  (
( A  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
11 ineq2 3523 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
12 fveq2 5714 . . . . . . 7  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1312ineq2d 3529 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  ( _|_ `  y
) )  =  ( A  i^i  ( _|_ `  B ) ) )
1411, 13oveq12d 6085 . . . . 5  |-  ( y  =  B  ->  (
( A  i^i  y
)  vH  ( A  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
1514eqeq2d 2441 . . . 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 692 . . 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 23068 . . 3  |-  C_H  =  { <. x ,  y
>.  |  ( (
x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
188, 16, 17brabg 4461 . 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 851 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3306   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   CHcch 22415   _|_cort 22416    vH chj 22419    C_H ccm 22422
This theorem is referenced by:  cmbri  23075  cm2j  23105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-iota 5404  df-fv 5448  df-ov 6070  df-cm 23068
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