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Theorem zlmval 17847
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 2979 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 oveq1 6097 . . . . 5  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,ring >. )  =  ( G sSet  <. (Scalar `  ndx ) ,ring >. ) )
4 fveq2 5688 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 zlmval.m . . . . . . 7  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2491 . . . . . 6  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76opeq2d 4063 . . . . 5  |-  ( g  =  G  ->  <. ( .s `  ndx ) ,  (.g `  g ) >.  =  <. ( .s `  ndx ) ,  .x.  >. )
83, 7oveq12d 6108 . . . 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 17836 . . . 4  |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g ) >.
) )
10 ovex 6115 . . . 4  |-  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. )  e.  _V
118, 9, 10fvmpt 5771 . . 3  |-  ( G  e.  _V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
122, 11syl 16 . 2  |-  ( G  e.  V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
131, 12syl5eq 2485 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 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880   ` cfv 5415  (class class class)co 6090   ndxcnx 14167   sSet csts 14168  Scalarcsca 14237   .scvsca 14238  .gcmg 15410  ℤringzring 17783   ZModczlm 17832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-zlm 17836
This theorem is referenced by:  zlmlem  17848  zlmsca  17852  zlmvsca  17853  zlmds  26313  zlmtset  26314
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