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Theorem crngocom 28813
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 28810 . . . 4  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
54simprbi 464 . . 3  |-  ( R  e. CRingOps  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
6 oveq1 6110 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
7 oveq2 6111 . . . . 5  |-  ( x  =  A  ->  (
y H x )  =  ( y H A ) )
86, 7eqeq12d 2457 . . . 4  |-  ( x  =  A  ->  (
( x H y )  =  ( y H x )  <->  ( A H y )  =  ( y H A ) ) )
9 oveq2 6111 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
10 oveq1 6110 . . . . 5  |-  ( y  =  B  ->  (
y H A )  =  ( B H A ) )
119, 10eqeq12d 2457 . . . 4  |-  ( y  =  B  ->  (
( A H y )  =  ( y H A )  <->  ( A H B )  =  ( B H A ) ) )
128, 11rspc2v 3091 . . 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 469 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  =  ( B H A ) )
14133impb 1183 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   ran crn 4853   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   RingOpscrngo 23874  CRingOpsccring 28807
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-1st 6589  df-2nd 6590  df-rngo 23875  df-com2 23910  df-crngo 28808
This theorem is referenced by:  crngm23  28814  crngohomfo  28818  isidlc  28827  dmncan2  28889
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