Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngm4 Structured version   Visualization version   Unicode version
Theorem crngm4 33802
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 1039 . . . . . 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 33801 . . . . . 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 493 . . . . 5 |- ( ( R e. CRingOps /\ ( ( A e. X /\ B e. X ) /\ C e. X ) ) -> ( ( A H B ) H C ) = ( ( A H C ) H B ) )
76adantrrr 761 . . . 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 6665 . . 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 33799 . . . 4 |- ( R e. CRingOps -> R e. RingOps )
102, 3, 4rngocl 33700 . . . . . . . 8 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
11103expb 1266 . . . . . . 7 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X ) ) -> ( A H B ) e. X )
1211adantrr 753 . . . . . 6 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) ) -> ( A H B ) e. X )
13 simprrl 804 . . . . . 6 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) ) -> C e. X )
14 simprrr 805 . . . . . 6 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) ) -> D e. X )
1512, 13, 143jca 1242 . . . . 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 33705 . . . . 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 487 . . . 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 488 . . 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 33700 . . . . . . . . 9 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
20193expb 1266 . . . . . . . 8 |- ( ( R e. RingOps /\ ( A e. X /\ C e. X ) ) -> ( A H C ) e. X )
2120adantrlr 759 . . . . . . 7 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ C e. X ) ) -> ( A H C ) e. X )
2221adantrrr 761 . . . . . 6 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) ) -> ( A H C ) e. X )
23 simprlr 803 . . . . . 6 |- ( ( R e. RingOps /\ ( ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) ) -> B e. X )
2422, 23, 143jca 1242 . . . . 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 33705 . . . . 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 487 . . . 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 488 . . 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 2664 . 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 1260 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693  CRingOpsccring 33792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694  df-com2 33789  df-crngo 33793
This theorem is referenced by:  ispridlc  33869
  Copyright terms: Public domain W3C validator