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Theorem crngocom 30228
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 30225 . . . 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 6292 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
7 oveq2 6293 . . . . 5  |-  ( x  =  A  ->  (
y H x )  =  ( y H A ) )
86, 7eqeq12d 2489 . . . 4  |-  ( x  =  A  ->  (
( x H y )  =  ( y H x )  <->  ( A H y )  =  ( y H A ) ) )
9 oveq2 6293 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
10 oveq1 6292 . . . . 5  |-  ( y  =  B  ->  (
y H A )  =  ( B H A ) )
119, 10eqeq12d 2489 . . . 4  |-  ( y  =  B  ->  (
( A H y )  =  ( y H A )  <->  ( A H B )  =  ( B H A ) ) )
128, 11rspc2v 3223 . . 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 1192 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   ran crn 5000   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   RingOpscrngo 25150  CRingOpsccring 30222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-1st 6785  df-2nd 6786  df-rngo 25151  df-com2 25186  df-crngo 30223
This theorem is referenced by:  crngm23  30229  crngohomfo  30233  isidlc  30242  dmncan2  30304
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