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Theorem rngodi 21965
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 21960 . . . 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 2773 . . . . 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 2773 . . . 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 2773 . . 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 6080 . . . . 5  |-  ( x  =  A  ->  (
x H ( y G z ) )  =  ( A H ( y G z ) ) )
12 oveq1 6080 . . . . . 6  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
13 oveq1 6080 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1412, 13oveq12d 6091 . . . . 5  |-  ( x  =  A  ->  (
( x H y ) G ( x H z ) )  =  ( ( A H y ) G ( A H z ) ) )
1511, 14eqeq12d 2449 . . . 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 6080 . . . . . 6  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1716oveq2d 6089 . . . . 5  |-  ( y  =  B  ->  ( A H ( y G z ) )  =  ( A H ( B G z ) ) )
18 oveq2 6081 . . . . . 6  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
1918oveq1d 6088 . . . . 5  |-  ( y  =  B  ->  (
( A H y ) G ( A H z ) )  =  ( ( A H B ) G ( A H z ) ) )
2017, 19eqeq12d 2449 . . . 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 6081 . . . . . 6  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
2221oveq2d 6089 . . . . 5  |-  ( z  =  C  ->  ( A H ( B G z ) )  =  ( A H ( B G C ) ) )
23 oveq2 6081 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
2423oveq2d 6089 . . . . 5  |-  ( z  =  C  ->  (
( A H B ) G ( A H z ) )  =  ( ( A H B ) G ( A H C ) ) )
2522, 24eqeq12d 2449 . . . 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 3053 . . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   AbelOpcablo 21861   RingOpscrngo 21955
This theorem is referenced by:  rngorz  21982  rngonegmn1r  26557  rngosubdi  26560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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