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Theorem crngm23 26627
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 26626 . . . 4  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
543adant3r1 1160 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
65oveq2d 5874 . 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 26625 . . 3  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
81, 2, 3rngoass 21054 . . 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 457 . 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 21054 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
11103exp2 1169 . . . . 5  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( C  e.  X  ->  ( B  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
1211com34 77 . . . 4  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
13123imp2 1166 . . 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 457 . 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 2325 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042  CRingOpsccring 26620
This theorem is referenced by:  crngm4  26628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043  df-com2 21078  df-crngo 26621
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