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Theorem crngm23 26593
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 26592 . . . 4  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
543adant3r1 1162 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
65oveq2d 6089 . 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 26591 . . 3  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
81, 2, 3rngoass 21967 . . 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 458 . 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 21967 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
11103exp2 1171 . . . . 5  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( C  e.  X  ->  ( B  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
1211com34 79 . . . 4  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
13123imp2 1168 . . 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 458 . 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 2477 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RingOpscrngo 21955  CRingOpsccring 26586
This theorem is referenced by:  crngm4  26594
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-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956  df-com2 21991  df-crngo 26587
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