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Theorem crngm23 29989
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 29988 . . . 4  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
543adant3r1 1200 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
65oveq2d 6291 . 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 29987 . . 3  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
81, 2, 3rngoass 25051 . . 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 25051 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
11103exp2 1209 . . . . 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 1206 . . 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 2511 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 968    = wceq 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   RingOpscrngo 25039  CRingOpsccring 29982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-1st 6774  df-2nd 6775  df-rngo 25040  df-com2 25075  df-crngo 29983
This theorem is referenced by:  crngm4  29990
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