Theorem lsmssv 18058

Description: Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmless2.v	|- B = ( Base ` G )
lsmless2.s	|- .(+) = ( LSSum ` G )

Assertion

Ref	Expression
lsmssv	|- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) C_ B )

Proof of Theorem lsmssv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsmless2.v	. . 3 |- B = ( Base ` G )
2		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
3		lsmless2.s	. . 3 |- .(+) = ( LSSum ` G )
4	1, 2, 3	lsmvalx 18054	. 2 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) )
5		simpl1 1064	. . . . . 6 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ ( x e. T /\ y e. U ) ) -> G e. Mnd )
6		simp2 1062	. . . . . . . 8 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> T C_ B )
7	6	sselda 3603	. . . . . . 7 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ x e. T ) -> x e. B )
8	7	adantrr 753	. . . . . 6 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ ( x e. T /\ y e. U ) ) -> x e. B )
9		simp3 1063	. . . . . . . 8 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> U C_ B )
10	9	sselda 3603	. . . . . . 7 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ y e. U ) -> y e. B )
11	10	adantrl 752	. . . . . 6 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ ( x e. T /\ y e. U ) ) -> y e. B )
12	1, 2	mndcl 17301	. . . . . 6 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
13	5, 8, 11, 12	syl3anc 1326	. . . . 5 |- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ ( x e. T /\ y e. U ) ) -> ( x ( +g ` G ) y ) e. B )
14	13	ralrimivva 2971	. . . 4 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> A. x e. T A. y e. U ( x ( +g ` G ) y ) e. B )
15		eqid 2622	. . . . 5 |- ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) = ( x e. T , y e. U |-> ( x ( +g ` G ) y ) )
16	15	fmpt2 7237	. . . 4 |- ( A. x e. T A. y e. U ( x ( +g ` G ) y ) e. B <-> ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) : ( T X. U ) --> B )
17	14, 16	sylib 208	. . 3 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) : ( T X. U ) --> B )
18		frn 6053	. . 3 |- ( ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) : ( T X. U ) --> B -> ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) C_ B )
19	17, 18	syl 17	. 2 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) C_ B )
20	4, 19	eqsstrd 3639	1 |- ( ( G e. Mnd /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) C_ B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294   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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-lsm 18051

This theorem is referenced by:  lsmsubm  18068  lsmass  18083  lsmcntzr  18093