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Theorem crngocom 31938
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 31935 . . . 4  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
54simprbi 465 . . 3  |-  ( R  e. CRingOps  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
6 oveq1 6312 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
7 oveq2 6313 . . . . 5  |-  ( x  =  A  ->  (
y H x )  =  ( y H A ) )
86, 7eqeq12d 2451 . . . 4  |-  ( x  =  A  ->  (
( x H y )  =  ( y H x )  <->  ( A H y )  =  ( y H A ) ) )
9 oveq2 6313 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
10 oveq1 6312 . . . . 5  |-  ( y  =  B  ->  (
y H A )  =  ( B H A ) )
119, 10eqeq12d 2451 . . . 4  |-  ( y  =  B  ->  (
( A H y )  =  ( y H A )  <->  ( A H B )  =  ( B H A ) ) )
128, 11rspc2v 3197 . . 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 471 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  =  ( B H A ) )
14133impb 1201 1  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   ran crn 4855   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   RingOpscrngo 25948  CRingOpsccring 31932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-1st 6807  df-2nd 6808  df-rngo 25949  df-com2 25984  df-crngo 31933
This theorem is referenced by:  crngm23  31939  crngohomfo  31943  isidlc  31952  dmncan2  32014
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