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Theorem crngm23 28755
Description: Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
crngm.1  |-  G  =  ( 1st `  R
)
crngm.2  |-  H  =  ( 2nd `  R
)
crngm.3  |-  X  =  ran  G
Assertion
Ref Expression
crngm23  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )

Proof of Theorem crngm23
StepHypRef Expression
1 crngm.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 crngm.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 crngm.3 . . . . 5  |-  X  =  ran  G
41, 2, 3crngocom 28754 . . . 4  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
543adant3r1 1196 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
65oveq2d 6102 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B H C ) )  =  ( A H ( C H B ) ) )
7 crngorngo 28753 . . 3  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
81, 2, 3rngoass 23825 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
97, 8sylan 471 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
101, 2, 3rngoass 23825 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
11103exp2 1205 . . . . 5  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( C  e.  X  ->  ( B  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
1211com34 83 . . . 4  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
13123imp2 1202 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
147, 13sylan 471 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
156, 9, 143eqtr4d 2480 1  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4836   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   RingOpscrngo 23813  CRingOpsccring 28748
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-1st 6572  df-2nd 6573  df-rngo 23814  df-com2 23849  df-crngo 28749
This theorem is referenced by:  crngm4  28756
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