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Theorem rngoass 21107
Description: Associative 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
rngoass  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )

Proof of Theorem rngoass
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 ringi.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . . . . 6  |-  X  =  ran  G
41, 2, 3rngoi 21100 . . . . 5  |-  ( 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 . . . 4  |-  ( 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 . . 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 ) ) ) )
7 simp1 955 . . . . . 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 ) H z )  =  ( x H ( y H z ) ) )
87ralimi 2652 . . . . 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 ) H z )  =  ( x H ( y H z ) ) )
98ralimi 2652 . . . 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 ) H z )  =  ( x H ( y H z ) ) )
109ralimi 2652 . . 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 ) H z )  =  ( x H ( y H z ) ) )
116, 10syl 15 . 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 ) ) )
12 oveq1 5907 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
1312oveq1d 5915 . . . 4  |-  ( x  =  A  ->  (
( x H y ) H z )  =  ( ( A H y ) H z ) )
14 oveq1 5907 . . . 4  |-  ( x  =  A  ->  (
x H ( y H z ) )  =  ( A H ( y H z ) ) )
1513, 14eqeq12d 2330 . . 3  |-  ( x  =  A  ->  (
( ( x H y ) H z )  =  ( x H ( y H z ) )  <->  ( ( A H y ) H z )  =  ( A H ( y H z ) ) ) )
16 oveq2 5908 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
1716oveq1d 5915 . . . 4  |-  ( y  =  B  ->  (
( A H y ) H z )  =  ( ( A H B ) H z ) )
18 oveq1 5907 . . . . 5  |-  ( y  =  B  ->  (
y H z )  =  ( B H z ) )
1918oveq2d 5916 . . . 4  |-  ( y  =  B  ->  ( A H ( y H z ) )  =  ( A H ( B H z ) ) )
2017, 19eqeq12d 2330 . . 3  |-  ( y  =  B  ->  (
( ( A H y ) H z )  =  ( A H ( y H z ) )  <->  ( ( A H B ) H z )  =  ( A H ( B H z ) ) ) )
21 oveq2 5908 . . . 4  |-  ( z  =  C  ->  (
( A H B ) H z )  =  ( ( A H B ) H C ) )
22 oveq2 5908 . . . . 5  |-  ( z  =  C  ->  ( B H z )  =  ( B H C ) )
2322oveq2d 5916 . . . 4  |-  ( z  =  C  ->  ( A H ( B H z ) )  =  ( A H ( B H C ) ) )
2421, 23eqeq12d 2330 . . 3  |-  ( z  =  C  ->  (
( ( A H B ) H z )  =  ( A H ( B H z ) )  <->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) ) )
2515, 20, 24rspc3v 2927 . 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 ) )  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) ) )
2611, 25mpan9 455 1  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   E.wrex 2578    X. cxp 4724   ran crn 4727   -->wf 5288   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   AbelOpcablo 21001   RingOpscrngo 21095
This theorem is referenced by:  rngomndo  21141  zerdivemp1  21154  rngoneglmul  25730  rngonegrmul  25731  zerdivemp1x  25734  isdrngo2  25737  crngm23  25775  crngm4  25776  prnc  25840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-rngo 21096
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