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Theorem crngocom 26602
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 26599 . . . 4  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
54simprbi 451 . . 3  |-  ( R  e. CRingOps  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
6 oveq1 6080 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
7 oveq2 6081 . . . . 5  |-  ( x  =  A  ->  (
y H x )  =  ( y H A ) )
86, 7eqeq12d 2449 . . . 4  |-  ( x  =  A  ->  (
( x H y )  =  ( y H x )  <->  ( A H y )  =  ( y H A ) ) )
9 oveq2 6081 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
10 oveq1 6080 . . . . 5  |-  ( y  =  B  ->  (
y H A )  =  ( B H A ) )
119, 10eqeq12d 2449 . . . 4  |-  ( y  =  B  ->  (
( A H y )  =  ( y H A )  <->  ( A H B )  =  ( B H A ) ) )
128, 11rspc2v 3050 . . 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 456 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  =  ( B H A ) )
14133impb 1149 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RingOpscrngo 21955  CRingOpsccring 26596
This theorem is referenced by:  crngm23  26603  crngohomfo  26607  isidlc  26616  dmncan2  26678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956  df-com2 21991  df-crngo 26597
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