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Theorem crngm4 30327
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 975 . . . . . 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 30326 . . . . . 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 476 . . . . 5  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( ( A H B ) H C )  =  ( ( A H C ) H B ) )
76adantrrr 724 . . . 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 6310 . . 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 30324 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
102, 3, 4rngocl 25207 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
11103expb 1197 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
1211adantrr 716 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H B )  e.  X
)
13 simprrl 763 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  C  e.  X
)
14 simprrr 764 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  D  e.  X
)
1512, 13, 143jca 1176 . . . . 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 25212 . . . . 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 470 . . . 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 471 . . 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 25207 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
20193expb 1197 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
2120adantrlr 722 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( A H C )  e.  X )
2221adantrrr 724 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H C )  e.  X
)
23 simprlr 762 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  B  e.  X
)
2422, 23, 143jca 1176 . . . . 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 25212 . . . . 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 470 . . . 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 471 . . 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 2516 . 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 1192 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   RingOpscrngo 25200  CRingOpsccring 30319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-rngo 25201  df-com2 25236  df-crngo 30320
This theorem is referenced by:  ispridlc  30394
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