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Theorem rngodir 24018
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 24012 . . . 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 990 . . . . . 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 2814 . . . . 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 2814 . . . 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 2814 . . 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 6200 . . . . . 6  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1211oveq1d 6208 . . . . 5  |-  ( x  =  A  ->  (
( x G y ) H z )  =  ( ( A G y ) H z ) )
13 oveq1 6200 . . . . . 6  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413oveq1d 6208 . . . . 5  |-  ( x  =  A  ->  (
( x H z ) G ( y H z ) )  =  ( ( A H z ) G ( y H z ) ) )
1512, 14eqeq12d 2473 . . . 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 6201 . . . . . 6  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1716oveq1d 6208 . . . . 5  |-  ( y  =  B  ->  (
( A G y ) H z )  =  ( ( A G B ) H z ) )
18 oveq1 6200 . . . . . 6  |-  ( y  =  B  ->  (
y H z )  =  ( B H z ) )
1918oveq2d 6209 . . . . 5  |-  ( y  =  B  ->  (
( A H z ) G ( y H z ) )  =  ( ( A H z ) G ( B H z ) ) )
2017, 19eqeq12d 2473 . . . 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 6201 . . . . 5  |-  ( z  =  C  ->  (
( A G B ) H z )  =  ( ( A G B ) H C ) )
22 oveq2 6201 . . . . . 6  |-  ( z  =  C  ->  ( A H z )  =  ( A H C ) )
23 oveq2 6201 . . . . . 6  |-  ( z  =  C  ->  ( B H z )  =  ( B H C ) )
2422, 23oveq12d 6211 . . . . 5  |-  ( z  =  C  ->  (
( A H z ) G ( B H z ) )  =  ( ( A H C ) G ( B H C ) ) )
2521, 24eqeq12d 2473 . . . 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 3182 . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    X. cxp 4939   ran crn 4942   -->wf 5515   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   AbelOpcablo 23913   RingOpscrngo 24007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-1st 6680  df-2nd 6681  df-rngo 24008
This theorem is referenced by:  rngo2  24020  rngolz  24033  rngonegmn1l  28896  rngosubdir  28901  prnc  29008
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