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Theorem crngocom 25774
Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
crngocom.1  |-  G  =  ( 1st `  R
)
crngocom.2  |-  H  =  ( 2nd `  R
)
crngocom.3  |-  X  =  ran  G
Assertion
Ref Expression
crngocom  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )

Proof of Theorem crngocom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngocom.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 crngocom.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 crngocom.3 . . . . 5  |-  X  =  ran  G
41, 2, 3iscrngo2 25771 . . . 4  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
54simprbi 450 . . 3  |-  ( R  e. CRingOps  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
6 oveq1 5907 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
7 oveq2 5908 . . . . 5  |-  ( x  =  A  ->  (
y H x )  =  ( y H A ) )
86, 7eqeq12d 2330 . . . 4  |-  ( x  =  A  ->  (
( x H y )  =  ( y H x )  <->  ( A H y )  =  ( y H A ) ) )
9 oveq2 5908 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
10 oveq1 5907 . . . . 5  |-  ( y  =  B  ->  (
y H A )  =  ( B H A ) )
119, 10eqeq12d 2330 . . . 4  |-  ( y  =  B  ->  (
( A H y )  =  ( y H A )  <->  ( A H B )  =  ( B H A ) ) )
128, 11rspc2v 2924 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x )  ->  ( A H B )  =  ( B H A ) ) )
135, 12mpan9 455 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  =  ( B H A ) )
14133impb 1147 1  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   ran crn 4727   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   RingOpscrngo 21095  CRingOpsccring 25768
This theorem is referenced by:  crngm23  25775  crngohomfo  25779  isidlc  25788  dmncan2  25850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-rngo 21096  df-com2 21131  df-crngo 25769
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