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Theorem rngodi 21068
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 21063 . . . 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 2631 . . . . 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 2631 . . . 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 2631 . . 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 5881 . . . . 5  |-  ( x  =  A  ->  (
x H ( y G z ) )  =  ( A H ( y G z ) ) )
12 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
13 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1412, 13oveq12d 5892 . . . . 5  |-  ( x  =  A  ->  (
( x H y ) G ( x H z ) )  =  ( ( A H y ) G ( A H z ) ) )
1511, 14eqeq12d 2310 . . . 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 5881 . . . . . 6  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1716oveq2d 5890 . . . . 5  |-  ( y  =  B  ->  ( A H ( y G z ) )  =  ( A H ( B G z ) ) )
18 oveq2 5882 . . . . . 6  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
1918oveq1d 5889 . . . . 5  |-  ( y  =  B  ->  (
( A H y ) G ( A H z ) )  =  ( ( A H B ) G ( A H z ) ) )
2017, 19eqeq12d 2310 . . . 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 5882 . . . . . 6  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
2221oveq2d 5890 . . . . 5  |-  ( z  =  C  ->  ( A H ( B G z ) )  =  ( A H ( B G C ) ) )
23 oveq2 5882 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
2423oveq2d 5890 . . . . 5  |-  ( z  =  C  ->  (
( A H B ) G ( A H z ) )  =  ( ( A H B ) G ( A H C ) ) )
2522, 24eqeq12d 2310 . . . 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 2906 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   AbelOpcablo 20964   RingOpscrngo 21058
This theorem is referenced by:  rngorz  21085  multinvb  25526  glmrngo  25585  rngonegmn1r  26684  rngosubdi  26687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059
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