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Theorem zlmval 16752
Description: Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
zlmval.w  |-  W  =  ( ZMod `  G
)
zlmval.z  |-  Z  =  (flds  ZZ )
zlmval.m  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
zlmval  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) 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 2924 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 zlmval.z . . . . . . . 8  |-  Z  =  (flds  ZZ )
43opeq2i 3948 . . . . . . 7  |-  <. (Scalar ` 
ndx ) ,  Z >.  =  <. (Scalar `  ndx ) ,  (flds  ZZ ) >.
54oveq2i 6051 . . . . . 6  |-  ( g sSet  <. (Scalar `  ndx ) ,  Z >. )  =  ( g sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. )
6 oveq1 6047 . . . . . 6  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,  Z >. )  =  ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) )
75, 6syl5eqr 2450 . . . . 5  |-  ( g  =  G  ->  (
g sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. )  =  ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) )
8 fveq2 5687 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
9 zlmval.m . . . . . . 7  |-  .x.  =  (.g
`  G )
108, 9syl6eqr 2454 . . . . . 6  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
1110opeq2d 3951 . . . . 5  |-  ( g  =  G  ->  <. ( .s `  ndx ) ,  (.g `  g ) >.  =  <. ( .s `  ndx ) ,  .x.  >. )
127, 11oveq12d 6058 . . . 4  |-  ( g  =  G  ->  (
( g sSet  <. (Scalar ` 
ndx ) ,  (flds  ZZ )
>. ) sSet  <. ( .s
`  ndx ) ,  (.g `  g ) >. )  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
13 df-zlm 16738 . . . 4  |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,  (flds  ZZ )
>. ) sSet  <. ( .s
`  ndx ) ,  (.g `  g ) >. )
)
14 ovex 6065 . . . 4  |-  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. )  e.  _V
1512, 13, 14fvmpt 5765 . . 3  |-  ( G  e.  _V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. )
)
162, 15syl 16 . 2  |-  ( G  e.  V  ->  ( ZMod `  G )  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. )
)
171, 16syl5eq 2448 1  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s `  ndx ) , 
.x.  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   ` cfv 5413  (class class class)co 6040   ZZcz 10238   ndxcnx 13421   sSet csts 13422   ↾s cress 13425  Scalarcsca 13487   .scvsca 13488  .gcmg 14644  ℂfldccnfld 16658   ZModczlm 16734
This theorem is referenced by:  zlmlem  16753  zlmsca  16757  zlmvsca  16758  zlmds  24301  zlmtset  24302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-zlm 16738
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