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Theorem rngodir 21364
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
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 21358 . . . 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 simp3 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 G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )
87ralimi 2703 . . . . 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 G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )
98ralimi 2703 . . . 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 G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )
109ralimi 2703 . . 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 ) ) )
11 oveq1 5988 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1211oveq1d 5996 . . . . 5  |-  ( x  =  A  ->  (
( x G y ) H z )  =  ( ( A G y ) H z ) )
13 oveq1 5988 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413oveq1d 5996 . . . . 5  |-  ( x  =  A  ->  (
( x H z ) G ( y H z ) )  =  ( ( A H z ) G ( y H z ) ) )
1512, 14eqeq12d 2380 . . . 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 ) ) ) )
16 oveq2 5989 . . . . . 6  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1716oveq1d 5996 . . . . 5  |-  ( y  =  B  ->  (
( A G y ) H z )  =  ( ( A G B ) H z ) )
18 oveq1 5988 . . . . . 6  |-  ( y  =  B  ->  (
y H z )  =  ( B H z ) )
1918oveq2d 5997 . . . . 5  |-  ( y  =  B  ->  (
( A H z ) G ( y H z ) )  =  ( ( A H z ) G ( B H z ) ) )
2017, 19eqeq12d 2380 . . . 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 ) ) ) )
21 oveq2 5989 . . . . 5  |-  ( z  =  C  ->  (
( A G B ) H z )  =  ( ( A G B ) H C ) )
22 oveq2 5989 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
23 oveq2 5989 . . . . . 6  |-  ( z  =  C  ->  ( B H z )  =  ( B H C ) )
2422, 23oveq12d 5999 . . . . 5  |-  ( z  =  C  ->  (
( A H z ) G ( B H z ) )  =  ( ( A H C ) G ( B H C ) ) )
2521, 24eqeq12d 2380 . . . 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 ) ) ) )
2615, 20, 25rspc3v 2978 . . 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 ) ) ) )
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 G B ) H C )  =  ( ( A H C ) G ( B H C ) ) ) )
286, 27mpan9 455 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    X. cxp 4790   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   AbelOpcablo 21259   RingOpscrngo 21353
This theorem is referenced by:  rngo2  21366  rngolz  21379  rngonegmn1l  26086  rngosubdir  26091  prnc  26198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-1st 6249  df-2nd 6250  df-rngo 21354
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