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Theorem rngodi 21814
Description: Distributive law for the multiplication operation of a ring. (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
rngodi  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) )

Proof of Theorem rngodi
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 21809 . . . 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 450 . . 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 446 . 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 simp2 958 . . . . . 6  |-  ( ( ( ( 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 H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
87ralimi 2717 . . . . 5  |-  ( 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 H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
98ralimi 2717 . . . 4  |-  ( 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. y  e.  X  A. z  e.  X  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
109ralimi 2717 . . 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 H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
11 oveq1 6020 . . . . 5  |-  ( x  =  A  ->  (
x H ( y G z ) )  =  ( A H ( y G z ) ) )
12 oveq1 6020 . . . . . 6  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
13 oveq1 6020 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1412, 13oveq12d 6031 . . . . 5  |-  ( x  =  A  ->  (
( x H y ) G ( x H z ) )  =  ( ( A H y ) G ( A H z ) ) )
1511, 14eqeq12d 2394 . . . 4  |-  ( x  =  A  ->  (
( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  <->  ( A H ( y G z ) )  =  ( ( A H y ) G ( A H z ) ) ) )
16 oveq1 6020 . . . . . 6  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1716oveq2d 6029 . . . . 5  |-  ( y  =  B  ->  ( A H ( y G z ) )  =  ( A H ( B G z ) ) )
18 oveq2 6021 . . . . . 6  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
1918oveq1d 6028 . . . . 5  |-  ( y  =  B  ->  (
( A H y ) G ( A H z ) )  =  ( ( A H B ) G ( A H z ) ) )
2017, 19eqeq12d 2394 . . . 4  |-  ( y  =  B  ->  (
( A H ( y G z ) )  =  ( ( A H y ) G ( A H z ) )  <->  ( A H ( B G z ) )  =  ( ( A H B ) G ( A H z ) ) ) )
21 oveq2 6021 . . . . . 6  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
2221oveq2d 6029 . . . . 5  |-  ( z  =  C  ->  ( A H ( B G z ) )  =  ( A H ( B G C ) ) )
23 oveq2 6021 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
2423oveq2d 6029 . . . . 5  |-  ( z  =  C  ->  (
( A H B ) G ( A H z ) )  =  ( ( A H B ) G ( A H C ) ) )
2522, 24eqeq12d 2394 . . . 4  |-  ( z  =  C  ->  (
( A H ( B G z ) )  =  ( ( A H B ) G ( A H z ) )  <->  ( A H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) ) )
2615, 20, 25rspc3v 2997 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  ->  ( A H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) ) )
2710, 26syl5 30 . 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 H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) ) )
286, 27mpan9 456 1  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    X. cxp 4809   ran crn 4812   -->wf 5383   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   AbelOpcablo 21710   RingOpscrngo 21804
This theorem is referenced by:  rngorz  21831  rngonegmn1r  26250  rngosubdi  26253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-1st 6281  df-2nd 6282  df-rngo 21805
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