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Theorem zlmval 19018
Description: Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
Hypotheses
Ref Expression
zlmval.w  |-  W  =  ( ZMod `  G
)
zlmval.m  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
zlmval  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. )
)

Proof of Theorem zlmval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 zlmval.w . 2  |-  W  =  ( ZMod `  G
)
2 elex 3096 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 oveq1 6312 . . . . 5  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,ring >. )  =  ( G sSet  <. (Scalar `  ndx ) ,ring >. ) )
4 fveq2 5881 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 zlmval.m . . . . . . 7  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2488 . . . . . 6  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76opeq2d 4197 . . . . 5  |-  ( g  =  G  ->  <. ( .s `  ndx ) ,  (.g `  g ) >.  =  <. ( .s `  ndx ) ,  .x.  >. )
83, 7oveq12d 6323 . . . 4  |-  ( g  =  G  ->  (
( g sSet  <. (Scalar ` 
ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g ) >.
)  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
9 df-zlm 19007 . . . 4  |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g ) >.
) )
10 ovex 6333 . . . 4  |-  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. )  e.  _V
118, 9, 10fvmpt 5964 . . 3  |-  ( G  e.  _V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
122, 11syl 17 . 2  |-  ( G  e.  V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
131, 12syl5eq 2482 1  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   <.cop 4008   ` cfv 5601  (class class class)co 6305   ndxcnx 15081   sSet csts 15082  Scalarcsca 15155   .scvsca 15156  .gcmg 16623  ℤringzring 18973   ZModczlm 19003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-zlm 19007
This theorem is referenced by:  zlmlem  19019  zlmsca  19023  zlmvsca  19024  zlmds  28607  zlmtset  28608
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