Theorem subglsm 18086

Description: The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypotheses

Ref	Expression
subglsm.h	|- H = ( G ↾s S )
subglsm.s	|- .(+) = ( LSSum ` G )
subglsm.a	|- A = ( LSSum ` H )

Assertion

Ref	Expression
subglsm	|- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ( T .(+) U ) = ( T A U ) )

Proof of Theorem subglsm

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

Step	Hyp	Ref	Expression
1		simp11 1091	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) /\ x e. T /\ y e. U ) -> S e. (SubGrp ` G ) )
2		subglsm.h	. . . . . . 7 |- H = ( G ↾s S )
3		eqid 2622	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
4	2, 3	ressplusg 15993	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
5	1, 4	syl 17	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) /\ x e. T /\ y e. U ) -> ( +g ` G ) = ( +g ` H ) )
6	5	oveqd 6667	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) /\ x e. T /\ y e. U ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
7	6	mpt2eq3dva 6719	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) = ( x e. T , y e. U |-> ( x ( +g ` H ) y ) ) )
8	7	rneqd 5353	. 2 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) = ran ( x e. T , y e. U |-> ( x ( +g ` H ) y ) ) )
9		subgrcl 17599	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
10	9	3ad2ant1 1082	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> G e. Grp )
11		simp2 1062	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> T C_ S )
12		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
13	12	subgss 17595	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
14	13	3ad2ant1 1082	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> S C_ ( Base ` G ) )
15	11, 14	sstrd 3613	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> T C_ ( Base ` G ) )
16		simp3 1063	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> U C_ S )
17	16, 14	sstrd 3613	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> U C_ ( Base ` G ) )
18		subglsm.s	. . . 4 |- .(+) = ( LSSum ` G )
19	12, 3, 18	lsmvalx 18054	. . 3 |- ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) )
20	10, 15, 17, 19	syl3anc 1326	. 2 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ( T .(+) U ) = ran ( x e. T , y e. U |-> ( x ( +g ` G ) y ) ) )
21	2	subggrp 17597	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
22	21	3ad2ant1 1082	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> H e. Grp )
23	2	subgbas 17598	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
24	23	3ad2ant1 1082	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> S = ( Base ` H ) )
25	11, 24	sseqtrd 3641	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> T C_ ( Base ` H ) )
26	16, 24	sseqtrd 3641	. . 3 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> U C_ ( Base ` H ) )
27		eqid 2622	. . . 4 |- ( Base ` H ) = ( Base ` H )
28		eqid 2622	. . . 4 |- ( +g ` H ) = ( +g ` H )
29		subglsm.a	. . . 4 |- A = ( LSSum ` H )
30	27, 28, 29	lsmvalx 18054	. . 3 |- ( ( H e. Grp /\ T C_ ( Base ` H ) /\ U C_ ( Base ` H ) ) -> ( T A U ) = ran ( x e. T , y e. U |-> ( x ( +g ` H ) y ) ) )
31	22, 25, 26, 30	syl3anc 1326	. 2 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ( T A U ) = ran ( x e. T , y e. U |-> ( x ( +g ` H ) y ) ) )
32	8, 20, 31	3eqtr4d 2666	1 |- ( ( S e. (SubGrp ` G ) /\ T C_ S /\ U C_ S ) -> ( T .(+) U ) = ( T A U ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   ↾_s cress 15858   +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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-subg 17591  df-lsm 18051

This theorem is referenced by:  pgpfaclem1  18480