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Theorem crngm23 26730
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
crngm23  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )

Proof of Theorem crngm23
StepHypRef Expression
1 crngm.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 crngm.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 crngm.3 . . . . 5  |-  X  =  ran  G
41, 2, 3crngocom 26729 . . . 4  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
543adant3r1 1160 . . 3  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
65oveq2d 5890 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B H C ) )  =  ( A H ( C H B ) ) )
7 crngorngo 26728 . . 3  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
81, 2, 3rngoass 21070 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
97, 8sylan 457 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
101, 2, 3rngoass 21070 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
11103exp2 1169 . . . . 5  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( C  e.  X  ->  ( B  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
1211com34 77 . . . 4  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
( A H C ) H B )  =  ( A H ( C H B ) ) ) ) ) )
13123imp2 1166 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
147, 13sylan 457 . 2  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) H B )  =  ( A H ( C H B ) ) )
156, 9, 143eqtr4d 2338 1  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
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:  crngm4  26731
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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