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Theorem cmbr 24985
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 2501 . . . . 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 3543 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
5 ineq1 3543 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  ( _|_ `  y ) )  =  ( A  i^i  ( _|_ `  y ) ) )
64, 5oveq12d 6107 . . . . 5  |-  ( x  =  A  ->  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  y )  vH  ( A  i^i  ( _|_ `  y
) ) ) )
73, 6eqeq12d 2455 . . . 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 2501 . . . . 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 3544 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
12 fveq2 5689 . . . . . . 7  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1312ineq2d 3550 . . . . . 6  |-  ( y  =  B  ->  ( A  i^i  ( _|_ `  y
) )  =  ( A  i^i  ( _|_ `  B ) ) )
1411, 13oveq12d 6107 . . . . 5  |-  ( y  =  B  ->  (
( A  i^i  y
)  vH  ( A  i^i  ( _|_ `  y
) ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
1514eqeq2d 2452 . . . 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 24984 . . 3  |-  C_H  =  { <. x ,  y
>.  |  ( (
x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
188, 16, 17brabg 4606 . 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 875 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 1369    e. wcel 1756    i^i cin 3325   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CHcch 24329   _|_cort 24330    vH chj 24333    C_H ccm 24336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-iota 5379  df-fv 5424  df-ov 6092  df-cm 24984
This theorem is referenced by:  cmbri  24991  cm2j  25021
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