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Theorem crngm4 30375
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 976 . . . . . 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 30374 . . . . . 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 6296 . . 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 30372 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
102, 3, 4rngocl 25360 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
11103expb 1198 . . . . . . 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 765 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  C  e.  X
)
14 simprrr 766 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  D  e.  X
)
1512, 13, 143jca 1177 . . . . 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 25365 . . . . 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 25360 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
20193expb 1198 . . . . . . . 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 764 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  B  e.  X
)
2422, 23, 143jca 1177 . . . . 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 25365 . . . . 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 2492 . 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 1193 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 974    = wceq 1383    e. wcel 1804   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   RingOpscrngo 25353  CRingOpsccring 30367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786  df-rngo 25354  df-com2 25389  df-crngo 30368
This theorem is referenced by:  ispridlc  30442
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