Theorem lsmelval 18064

Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmelval.a	|- .+ = ( +g ` G )
lsmelval.p	|- .(+) = ( LSSum ` G )

Assertion

Ref	Expression
lsmelval	|- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )

Distinct variable groups:   y, z, .+   y, T, z   y, U, z   y, G, z   y, X, z

Allowed substitution hints:   .(+) ( y, z)

Proof of Theorem lsmelval

Step	Hyp	Ref	Expression
1		subgrcl 17599	. . 3 |- ( T e. (SubGrp ` G ) -> G e. Grp )
2	1	adantr 481	. 2 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> G e. Grp )
3		eqid 2622	. . . 4 |- ( Base ` G ) = ( Base ` G )
4	3	subgss 17595	. . 3 |- ( T e. (SubGrp ` G ) -> T C_ ( Base ` G ) )
5	4	adantr 481	. 2 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> T C_ ( Base ` G ) )
6	3	subgss 17595	. . 3 |- ( U e. (SubGrp ` G ) -> U C_ ( Base ` G ) )
7	6	adantl 482	. 2 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> U C_ ( Base ` G ) )
8		lsmelval.a	. . 3 |- .+ = ( +g ` G )
9		lsmelval.p	. . 3 |- .(+) = ( LSSum ` G )
10	3, 8, 9	lsmelvalx 18055	. 2 |- ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )
11	2, 5, 7, 10	syl3anc 1326	1 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422  SubGrpcsubg 17588   LSSumclsm 18049

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-subg 17591  df-lsm 18051

This theorem is referenced by:  lsmelvalm  18066  lsmsubg  18069  lsmcom2  18070  lsmmod  18088  lsmdisj2  18095  pj1eu  18109  lsmcl  19083  lsmspsn  19084  lsmelval2  19085  lsmcv  19141  lsmsat  34295  lshpsmreu  34396  dvhopellsm  36406  diblsmopel  36460  cdlemn11c  36498  dihord11c  36513  hdmapglem7a  37219