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Theorem crngm4 26614
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
crngm4  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )

Proof of Theorem crngm4
StepHypRef Expression
1 df-3an 939 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( ( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )
2 crngm.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
3 crngm.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
4 crngm.3 . . . . . . 7  |-  X  =  ran  G
52, 3, 4crngm23 26613 . . . . . 6  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
61, 5sylan2br 464 . . . . 5  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( ( A H B ) H C )  =  ( ( A H C ) H B ) )
76adantrrr 707 . . . 4  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
87oveq1d 6097 . . 3  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( ( A H B ) H C ) H D )  =  ( ( ( A H C ) H B ) H D ) )
9 crngorngo 26611 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
102, 3, 4rngocl 21971 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
11103expb 1155 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
1211adantrr 699 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H B )  e.  X
)
13 simprrl 742 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  C  e.  X
)
14 simprrr 743 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  D  e.  X
)
1512, 13, 143jca 1135 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( A H B )  e.  X  /\  C  e.  X  /\  D  e.  X ) )
162, 3, 4rngoass 21976 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( A H B )  e.  X  /\  C  e.  X  /\  D  e.  X )
)  ->  ( (
( A H B ) H C ) H D )  =  ( ( A H B ) H ( C H D ) ) )
1715, 16syldan 458 . . . 4  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( ( A H B ) H C ) H D )  =  ( ( A H B ) H ( C H D ) ) )
189, 17sylan 459 . . 3  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( ( A H B ) H C ) H D )  =  ( ( A H B ) H ( C H D ) ) )
192, 3, 4rngocl 21971 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
20193expb 1155 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
2120adantrlr 705 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( A H C )  e.  X )
2221adantrrr 707 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H C )  e.  X
)
23 simprlr 741 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  B  e.  X
)
2422, 23, 143jca 1135 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( A H C )  e.  X  /\  B  e.  X  /\  D  e.  X ) )
252, 3, 4rngoass 21976 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( A H C )  e.  X  /\  B  e.  X  /\  D  e.  X )
)  ->  ( (
( A H C ) H B ) H D )  =  ( ( A H C ) H ( B H D ) ) )
2624, 25syldan 458 . . . 4  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( ( A H C ) H B ) H D )  =  ( ( A H C ) H ( B H D ) ) )
279, 26sylan 459 . . 3  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( ( A H C ) H B ) H D )  =  ( ( A H C ) H ( B H D ) ) )
288, 18, 273eqtr3d 2477 . 2  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )
29283impb 1150 1  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4880   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   RingOpscrngo 21964  CRingOpsccring 26606
This theorem is referenced by:  ispridlc  26681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-1st 6350  df-2nd 6351  df-rngo 21965  df-com2 22000  df-crngo 26607
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