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Theorem rngodir 25366
Description: Distributive law for the multiplication operation of a ring (right-distributivity). (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 25360 . . . 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 463 . . 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 459 . 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 999 . . . . . 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 2836 . . . . 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 2836 . . . 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 2836 . . 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 6288 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1211oveq1d 6296 . . . . 5  |-  ( x  =  A  ->  (
( x G y ) H z )  =  ( ( A G y ) H z ) )
13 oveq1 6288 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413oveq1d 6296 . . . . 5  |-  ( x  =  A  ->  (
( x H z ) G ( y H z ) )  =  ( ( A H z ) G ( y H z ) ) )
1512, 14eqeq12d 2465 . . . 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 6289 . . . . . 6  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1716oveq1d 6296 . . . . 5  |-  ( y  =  B  ->  (
( A G y ) H z )  =  ( ( A G B ) H z ) )
18 oveq1 6288 . . . . . 6  |-  ( y  =  B  ->  (
y H z )  =  ( B H z ) )
1918oveq2d 6297 . . . . 5  |-  ( y  =  B  ->  (
( A H z ) G ( y H z ) )  =  ( ( A H z ) G ( B H z ) ) )
2017, 19eqeq12d 2465 . . . 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 6289 . . . . 5  |-  ( z  =  C  ->  (
( A G B ) H z )  =  ( ( A G B ) H C ) )
22 oveq2 6289 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
23 oveq2 6289 . . . . . 6  |-  ( z  =  C  ->  ( B H z )  =  ( B H C ) )
2422, 23oveq12d 6299 . . . . 5  |-  ( z  =  C  ->  (
( A H z ) G ( B H z ) )  =  ( ( A H C ) G ( B H C ) ) )
2521, 24eqeq12d 2465 . . . 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 3208 . . 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 32 . 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 469 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    X. cxp 4987   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   AbelOpcablo 25261   RingOpscrngo 25355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786  df-rngo 25356
This theorem is referenced by:  rngo2  25368  rngolz  25381  rngonegmn1l  30328  rngosubdir  30333  prnc  30440
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