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Theorem crngm4 26731
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 936 . . . . . 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 26730 . . . . . 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 462 . . . . 5  |-  ( ( R  e. CRingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( ( A H B ) H C )  =  ( ( A H C ) H B ) )
76adantrrr 705 . . . 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 5889 . . 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 26728 . . . 4  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
102, 3, 4rngocl 21065 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
11103expb 1152 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
1211adantrr 697 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H B )  e.  X
)
13 simprrl 740 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  C  e.  X
)
14 simprrr 741 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  D  e.  X
)
1512, 13, 143jca 1132 . . . . 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 21070 . . . . 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 456 . . . 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 457 . . 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 21065 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
20193expb 1152 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
2120adantrlr 703 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( A H C )  e.  X )
2221adantrrr 705 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( A H C )  e.  X
)
23 simprlr 739 . . . . . 6  |-  ( ( R  e.  RingOps  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  B  e.  X
)
2422, 23, 143jca 1132 . . . . 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 21070 . . . . 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 456 . . . 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 457 . . 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 2336 . 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 1147 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   RingOpscrngo 21058  CRingOpsccring 26723
This theorem is referenced by:  ispridlc  26798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059  df-com2 21094  df-crngo 26724
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