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Theorem rngodi 21052
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 21047 . . . 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 449 . . 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 445 . 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 956 . . . . . 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 2618 . . . . 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 2618 . . . 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 2618 . . 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 5865 . . . . 5  |-  ( x  =  A  ->  (
x H ( y G z ) )  =  ( A H ( y G z ) ) )
12 oveq1 5865 . . . . . 6  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
13 oveq1 5865 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1412, 13oveq12d 5876 . . . . 5  |-  ( x  =  A  ->  (
( x H y ) G ( x H z ) )  =  ( ( A H y ) G ( A H z ) ) )
1511, 14eqeq12d 2297 . . . 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 5865 . . . . . 6  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1716oveq2d 5874 . . . . 5  |-  ( y  =  B  ->  ( A H ( y G z ) )  =  ( A H ( B G z ) ) )
18 oveq2 5866 . . . . . 6  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
1918oveq1d 5873 . . . . 5  |-  ( y  =  B  ->  (
( A H y ) G ( A H z ) )  =  ( ( A H B ) G ( A H z ) ) )
2017, 19eqeq12d 2297 . . . 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 5866 . . . . . 6  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
2221oveq2d 5874 . . . . 5  |-  ( z  =  C  ->  ( A H ( B G z ) )  =  ( A H ( B G C ) ) )
23 oveq2 5866 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
2423oveq2d 5874 . . . . 5  |-  ( z  =  C  ->  (
( A H B ) G ( A H z ) )  =  ( ( A H B ) G ( A H C ) ) )
2522, 24eqeq12d 2297 . . . 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 2893 . . 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 28 . 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 455 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   AbelOpcablo 20948   RingOpscrngo 21042
This theorem is referenced by:  rngorz  21069  multinvb  24835  glmrngo  24894  rngonegmn1r  25993  rngosubdi  25996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043
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