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Theorem rngodir 25092
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 25086 . . . 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 998 . . . . . 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 2857 . . . . 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 2857 . . . 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 2857 . . 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 6291 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1211oveq1d 6299 . . . . 5  |-  ( x  =  A  ->  (
( x G y ) H z )  =  ( ( A G y ) H z ) )
13 oveq1 6291 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413oveq1d 6299 . . . . 5  |-  ( x  =  A  ->  (
( x H z ) G ( y H z ) )  =  ( ( A H z ) G ( y H z ) ) )
1512, 14eqeq12d 2489 . . . 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 6292 . . . . . 6  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1716oveq1d 6299 . . . . 5  |-  ( y  =  B  ->  (
( A G y ) H z )  =  ( ( A G B ) H z ) )
18 oveq1 6291 . . . . . 6  |-  ( y  =  B  ->  (
y H z )  =  ( B H z ) )
1918oveq2d 6300 . . . . 5  |-  ( y  =  B  ->  (
( A H z ) G ( y H z ) )  =  ( ( A H z ) G ( B H z ) ) )
2017, 19eqeq12d 2489 . . . 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 6292 . . . . 5  |-  ( z  =  C  ->  (
( A G B ) H z )  =  ( ( A G B ) H C ) )
22 oveq2 6292 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
23 oveq2 6292 . . . . . 6  |-  ( z  =  C  ->  ( B H z )  =  ( B H C ) )
2422, 23oveq12d 6302 . . . . 5  |-  ( z  =  C  ->  (
( A H z ) G ( B H z ) )  =  ( ( A H C ) G ( B H C ) ) )
2521, 24eqeq12d 2489 . . . 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 3226 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    X. cxp 4997   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   AbelOpcablo 24987   RingOpscrngo 25081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-1st 6784  df-2nd 6785  df-rngo 25082
This theorem is referenced by:  rngo2  25094  rngolz  25107  rngonegmn1l  29983  rngosubdir  29988  prnc  30095
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