Theorem lsmelvali 18065

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
lsmelvali	|- ( ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )

Proof of Theorem lsmelvali

Step	Hyp	Ref	Expression
1		subgrcl 17599	. . . 4 |- ( T e. (SubGrp ` G ) -> G e. Grp )
2	1	adantr 481	. . 3 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> G e. Grp )
3		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
4	3	subgss 17595	. . . 4 |- ( T e. (SubGrp ` G ) -> T C_ ( Base ` G ) )
5	4	adantr 481	. . 3 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> T C_ ( Base ` G ) )
6	3	subgss 17595	. . . 4 |- ( U e. (SubGrp ` G ) -> U C_ ( Base ` G ) )
7	6	adantl 482	. . 3 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> U C_ ( Base ` G ) )
8	2, 5, 7	3jca 1242	. 2 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) )
9		lsmelval.a	. . 3 |- .+ = ( +g ` G )
10		lsmelval.p	. . 3 |- .(+) = ( LSSum ` G )
11	3, 9, 10	lsmelvalix 18056	. 2 |- ( ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )
12	8, 11	sylan 488	1 |- ( ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   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:  lsmsubg  18069  lsmmod  18088  lsmdisj2  18095  lsmhash  18118  ablfacrp  18465  lsmcl  19083  lsmelval2  19085  lsppreli  19090  lspprabs  19095  lspabs3  19121  pjthlem2  23209  lkrlsp  34389  dia2dimlem5  36357  mapdindp0  37008  hdmaprnlem3eN  37150