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Theorem rngodir 33704
Description: Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 |- G = ( 1st ` R )
ringi.2 |- H = ( 2nd ` R )
ringi.3 |- X = ran G
Assertion
Ref Expression
rngodir |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) H C ) = ( ( A H C ) G ( B H C ) ) )
Proof of Theorem rngodir
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5 |- G = ( 1st ` R )
2 ringi.2 . . . . 5 |- H = ( 2nd ` R )
3 ringi.3 . . . . 5 |- X = ran G
41, 2, 3rngoi 33698 . . . 4 |- ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) )
54simprd 479 . . 3 |- ( R e. RingOps -> ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) )
65simpld 475 . 2 |- ( R e. RingOps -> A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) )
7 simp3 1063 . . . . 5 |- ( ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) )
87ralimi 2952 . . . 4 |- ( A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> A. z e. X ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) )
982ralimi 2953 . . 3 |- ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> A. x e. X A. y e. X A. z e. X ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) )
10 oveq1 6657 . . . . . 6 |- ( x = A -> ( x G y ) = ( A G y ) )
1110oveq1d 6665 . . . . 5 |- ( x = A -> ( ( x G y ) H z ) = ( ( A G y ) H z ) )
12 oveq1 6657 . . . . . 6 |- ( x = A -> ( x H z ) = ( A H z ) )
1312oveq1d 6665 . . . . 5 |- ( x = A -> ( ( x H z ) G ( y H z ) ) = ( ( A H z ) G ( y H z ) ) )
1411, 13eqeq12d 2637 . . . 4 |- ( x = A -> ( ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) <-> ( ( A G y ) H z ) = ( ( A H z ) G ( y H z ) ) ) )
15 oveq2 6658 . . . . . 6 |- ( y = B -> ( A G y ) = ( A G B ) )
1615oveq1d 6665 . . . . 5 |- ( y = B -> ( ( A G y ) H z ) = ( ( A G B ) H z ) )
17 oveq1 6657 . . . . . 6 |- ( y = B -> ( y H z ) = ( B H z ) )
1817oveq2d 6666 . . . . 5 |- ( y = B -> ( ( A H z ) G ( y H z ) ) = ( ( A H z ) G ( B H z ) ) )
1916, 18eqeq12d 2637 . . . 4 |- ( y = B -> ( ( ( A G y ) H z ) = ( ( A H z ) G ( y H z ) ) <-> ( ( A G B ) H z ) = ( ( A H z ) G ( B H z ) ) ) )
20 oveq2 6658 . . . . 5 |- ( z = C -> ( ( A G B ) H z ) = ( ( A G B ) H C ) )
21 oveq2 6658 . . . . . 6 |- ( z = C -> ( A H z ) = ( A H C ) )
22 oveq2 6658 . . . . . 6 |- ( z = C -> ( B H z ) = ( B H C ) )
2321, 22oveq12d 6668 . . . . 5 |- ( z = C -> ( ( A H z ) G ( B H z ) ) = ( ( A H C ) G ( B H C ) ) )
2420, 23eqeq12d 2637 . . . 4 |- ( z = C -> ( ( ( A G B ) H z ) = ( ( A H z ) G ( B H z ) ) <-> ( ( A G B ) H C ) = ( ( A H C ) G ( B H C ) ) ) )
2514, 19, 24rspc3v 3325 . . 3 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( A. x e. X A. y e. X A. z e. X ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) -> ( ( A G B ) H C ) = ( ( A H C ) G ( B H C ) ) ) )
269, 25syl5 34 . 2 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> ( ( A G B ) H C ) = ( ( A H C ) G ( B H C ) ) ) )
276, 26mpan9 486 1 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) H C ) = ( ( A H C ) G ( B H C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   AbelOpcablo 27398   RingOpscrngo 33693
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
This theorem is referenced by:  rngo2  33706  rngolz  33721  rngonegmn1l  33740  rngosubdir  33745  prnc  33866
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