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Theorem crngm4 28829
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 967 . . . . . 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 28828 . . . . . 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 6127 . . 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 28826 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
102, 3, 4rngocl 23891 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
11103expb 1188 . . . . . . 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 1168 . . . . 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 23896 . . . . 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 23891 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
20193expb 1188 . . . . . . . 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 1168 . . . . 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 23896 . . . . 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 2483 . 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 1183 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 965    = wceq 1369    e. wcel 1756   ran crn 4862   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597   RingOpscrngo 23884  CRingOpsccring 28821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-1st 6598  df-2nd 6599  df-rngo 23885  df-com2 23920  df-crngo 28822
This theorem is referenced by:  ispridlc  28896
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