Theorem zlmval 19864

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 ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )

Proof of Theorem zlmval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zlmval.w	. 2 |- W = ( ZMod ` G )
2		elex 3212	. . 3 |- ( G e. V -> G e. _V )
3		oveq1 6657	. . . . 5 |- ( g = G -> ( g sSet <. (Scalar ` ndx ) , ℤring >. ) = ( G sSet <. (Scalar ` ndx ) , ℤring >. ) )
4		fveq2 6191	. . . . . . 7 |- ( g = G -> (.g ` g ) = (.g ` G ) )
5		zlmval.m	. . . . . . 7 |- .x. = (.g ` G )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( g = G -> (.g ` g ) = .x. )
7	6	opeq2d 4409	. . . . 5 |- ( g = G -> <. ( .s ` ndx ) , (.g ` g ) >. = <. ( .s ` ndx ) , .x. >. )
8	3, 7	oveq12d 6668	. . . 4 |- ( g = G -> ( ( g sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , (.g ` g ) >. ) = ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )
9		df-zlm 19853	. . . 4 |- ZMod = ( g e. _V |-> ( ( g sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , (.g ` g ) >. ) )
10		ovex 6678	. . . 4 |- ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) e. _V
11	8, 9, 10	fvmpt 6282	. . 3 |- ( G e. _V -> ( ZMod ` G ) = ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )
12	2, 11	syl 17	. 2 |- ( G e. V -> ( ZMod ` G ) = ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )
13	1, 12	syl5eq 2668	1 |- ( G e. V -> W = ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855  Scalarcsca 15944   .scvsca 15945  ._gcmg 17540  ℤ_ringzring 19818   ZModczlm 19849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-zlm 19853

This theorem is referenced by:  zlmlem  19865  zlmsca  19869  zlmvsca  19870  zlmds  30008  zlmtset  30009