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Theorem cmbr 22163
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 2343 . . . . 5  |-  ( x  =  A  ->  (
x  e.  CH  <->  A  e.  CH ) )
21anbi1d 685 . . . 4  |-  ( x  =  A  ->  (
( x  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  y  e.  CH )
) )
3 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 ineq1 3363 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
5 ineq1 3363 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  ( _|_ `  y ) )  =  ( A  i^i  ( _|_ `  y ) ) )
64, 5oveq12d 5876 . . . . 5  |-  ( x  =  A  ->  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) )
73, 6eqeq12d 2297 . . . 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 691 . . 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 2343 . . . . 5  |-  ( y  =  B  ->  (
y  e.  CH  <->  B  e.  CH ) )
109anbi2d 684 . . . 4  |-  ( y  =  B  ->  (
( A  e.  CH  /\  y  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
11 ineq2 3364 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
12 fveq2 5525 . . . . . . 7  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1312ineq2d 3370 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  ( _|_ `  y
) )  =  ( A  i^i  ( _|_ `  B ) ) )
1411, 13oveq12d 5876 . . . . 5  |-  ( y  =  B  ->  (
( A  i^i  y
)  vH  ( A  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
1514eqeq2d 2294 . . . 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 691 . . 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 22162 . . 3  |-  C_H  =  { <. x ,  y
>.  |  ( (
x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
188, 16, 17brabg 4284 . 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 850 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CHcch 21509   _|_cort 21510    vH chj 21513    C_H ccm 21516
This theorem is referenced by:  cmbri  22169  cm2j  22199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-cm 22162
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