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Theorem zlmval 18336
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 3122 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 oveq1 6290 . . . . 5  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,ring >. )  =  ( G sSet  <. (Scalar `  ndx ) ,ring >. ) )
4 fveq2 5865 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 zlmval.m . . . . . . 7  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2526 . . . . . 6  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76opeq2d 4220 . . . . 5  |-  ( g  =  G  ->  <. ( .s `  ndx ) ,  (.g `  g ) >.  =  <. ( .s `  ndx ) ,  .x.  >. )
83, 7oveq12d 6301 . . . 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 18325 . . . 4  |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g ) >.
) )
10 ovex 6308 . . . 4  |-  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. )  e.  _V
118, 9, 10fvmpt 5949 . . 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 2520 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 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   ` cfv 5587  (class class class)co 6283   ndxcnx 14486   sSet csts 14487  Scalarcsca 14557   .scvsca 14558  .gcmg 15730  ℤringzring 18272   ZModczlm 18321
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-zlm 18325
This theorem is referenced by:  zlmlem  18337  zlmsca  18341  zlmvsca  18342  zlmds  27597  zlmtset  27598
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