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Theorem zlmval 17947
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 2981 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 oveq1 6098 . . . . 5  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,ring >. )  =  ( G sSet  <. (Scalar `  ndx ) ,ring >. ) )
4 fveq2 5691 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 zlmval.m . . . . . . 7  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2493 . . . . . 6  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76opeq2d 4066 . . . . 5  |-  ( g  =  G  ->  <. ( .s `  ndx ) ,  (.g `  g ) >.  =  <. ( .s `  ndx ) ,  .x.  >. )
83, 7oveq12d 6109 . . . 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 17936 . . . 4  |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g ) >.
) )
10 ovex 6116 . . . 4  |-  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. )  e.  _V
118, 9, 10fvmpt 5774 . . 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 2487 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 1369    e. wcel 1756   _Vcvv 2972   <.cop 3883   ` cfv 5418  (class class class)co 6091   ndxcnx 14171   sSet csts 14172  Scalarcsca 14241   .scvsca 14242  .gcmg 15414  ℤringzring 17883   ZModczlm 17932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-zlm 17936
This theorem is referenced by:  zlmlem  17948  zlmsca  17952  zlmvsca  17953  zlmds  26393  zlmtset  26394
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