Theorem lsmmod2 18089

Description: Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypothesis

Ref	Expression
lsmmod.p	|- .(+) = ( LSSum ` G )

Assertion

Ref	Expression
lsmmod2	|- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> ( S i^i ( T .(+) U ) ) = ( ( S i^i T ) .(+) U ) )

Proof of Theorem lsmmod2

Step	Hyp	Ref	Expression
1		simpl3 1066	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> U e. (SubGrp ` G ) )
2		eqid 2622	. . . . . . 7 |- (oppg ` G ) = (oppg ` G )
3	2	oppgsubg 17793	. . . . . 6 |- (SubGrp ` G ) = (SubGrp ` (oppg ` G ) )
4	1, 3	syl6eleq 2711	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> U e. (SubGrp ` (oppg ` G ) ) )
5		simpl2 1065	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> T e. (SubGrp ` G ) )
6	5, 3	syl6eleq 2711	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> T e. (SubGrp ` (oppg ` G ) ) )
7		simpl1 1064	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> S e. (SubGrp ` G ) )
8	7, 3	syl6eleq 2711	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> S e. (SubGrp ` (oppg ` G ) ) )
9		simpr 477	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> U C_ S )
10		eqid 2622	. . . . . 6 |- ( LSSum ` (oppg ` G ) ) = ( LSSum ` (oppg ` G ) )
11	10	lsmmod 18088	. . . . 5 |- ( ( ( U e. (SubGrp ` (oppg ` G ) ) /\ T e. (SubGrp ` (oppg ` G ) ) /\ S e. (SubGrp ` (oppg ` G ) ) ) /\ U C_ S ) -> ( U ( LSSum ` (oppg ` G ) ) ( T i^i S ) ) = ( ( U ( LSSum ` (oppg ` G ) ) T ) i^i S ) )
12	4, 6, 8, 9, 11	syl31anc 1329	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> ( U ( LSSum ` (oppg ` G ) ) ( T i^i S ) ) = ( ( U ( LSSum ` (oppg ` G ) ) T ) i^i S ) )
13	12	eqcomd 2628	. . 3 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> ( ( U ( LSSum ` (oppg ` G ) ) T ) i^i S ) = ( U ( LSSum ` (oppg ` G ) ) ( T i^i S ) ) )
14		incom 3805	. . 3 |- ( ( U ( LSSum ` (oppg ` G ) ) T ) i^i S ) = ( S i^i ( U ( LSSum ` (oppg ` G ) ) T ) )
15		incom 3805	. . . 4 |- ( T i^i S ) = ( S i^i T )
16	15	oveq2i 6661	. . 3 |- ( U ( LSSum ` (oppg ` G ) ) ( T i^i S ) ) = ( U ( LSSum ` (oppg ` G ) ) ( S i^i T ) )
17	13, 14, 16	3eqtr3g 2679	. 2 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> ( S i^i ( U ( LSSum ` (oppg ` G ) ) T ) ) = ( U ( LSSum ` (oppg ` G ) ) ( S i^i T ) ) )
18		lsmmod.p	. . . 4 |- .(+) = ( LSSum ` G )
19	2, 18	oppglsm 18057	. . 3 |- ( U ( LSSum ` (oppg ` G ) ) T ) = ( T .(+) U )
20	19	ineq2i 3811	. 2 |- ( S i^i ( U ( LSSum ` (oppg ` G ) ) T ) ) = ( S i^i ( T .(+) U ) )
21	2, 18	oppglsm 18057	. 2 |- ( U ( LSSum ` (oppg ` G ) ) ( S i^i T ) ) = ( ( S i^i T ) .(+) U )
22	17, 20, 21	3eqtr3g 2679	1 |- ( ( ( S e. (SubGrp ` G ) /\ T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ U C_ S ) -> ( S i^i ( T .(+) U ) ) = ( ( S i^i T ) .(+) U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650  SubGrpcsubg 17588  opp_gcoppg 17775   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-rmo 2920  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-int 4476  df-iun 4522  df-iin 4523  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-oppg 17776  df-lsm 18051

This theorem is referenced by:  lcvexchlem3  34323  lcfrlem23  36854