Theorem subgmulg 17608

Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
subgmulgcl.t	|- .x. = (.g ` G )
subgmulg.h	|- H = ( G ↾s S )
subgmulg.t	|- .xb = (.g ` H )

Assertion

Ref	Expression
subgmulg	|- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Proof of Theorem subgmulg

Step	Hyp	Ref	Expression
1		subgmulg.h	. . . . . 6 |- H = ( G ↾s S )
2		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	subg0 17600	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
4	3	3ad2ant1 1082	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( 0g ` G ) = ( 0g ` H ) )
5	4	ifeq1d 4104	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
6		eqid 2622	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` G )
7	1, 6	ressplusg 15993	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
8	7	3ad2ant1 1082	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( +g ` G ) = ( +g ` H ) )
9	8	seqeq2d 12808	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
10	9	adantr 481	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
11	10	fveq1d 6193	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
12	11	ifeq1d 4104	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) )
13		simp2 1062	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. ZZ )
14	13	zred 11482	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. RR )
15		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
16		axlttri 10109	. . . . . . . . . . . 12 |- ( ( N e. RR /\ 0 e. RR ) -> ( N < 0 <-> -. ( N = 0 \/ 0 < N ) ) )
17	14, 15, 16	sylancl 694	. . . . . . . . . . 11 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 <-> -. ( N = 0 \/ 0 < N ) ) )
18		ioran 511	. . . . . . . . . . 11 |- ( -. ( N = 0 \/ 0 < N ) <-> ( -. N = 0 /\ -. 0 < N ) )
19	17, 18	syl6bb 276	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 <-> ( -. N = 0 /\ -. 0 < N ) ) )
20	19	biimpar 502	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> N < 0 )
21		simpl1 1064	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> S e. (SubGrp ` G ) )
22	13	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> N e. ZZ )
23	22	znegcld 11484	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. ZZ )
24	14	lt0neg1d 10597	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 <-> 0 < -u N ) )
25	24	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> 0 < -u N )
26		elnnz 11387	. . . . . . . . . . . . 13 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
27	23, 25, 26	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. NN )
28		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( Base ` G ) = ( Base ` G )
29	28	subgss 17595	. . . . . . . . . . . . . . 15 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
30	29	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S C_ ( Base ` G ) )
31		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. S )
32	30, 31	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` G ) )
33	32	adantr 481	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. ( Base ` G ) )
34		subgmulgcl.t	. . . . . . . . . . . . 13 |- .x. = (.g ` G )
35		eqid 2622	. . . . . . . . . . . . 13 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
36	28, 6, 34, 35	mulgnn 17547	. . . . . . . . . . . 12 |- ( ( -u N e. NN /\ X e. ( Base ` G ) ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
37	27, 33, 36	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
38	31	adantr 481	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. S )
39	34	subgmulgcl 17607	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ -u N e. ZZ /\ X e. S ) -> ( -u N .x. X ) e. S )
40	21, 23, 38, 39	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) e. S )
41	37, 40	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S )
42		eqid 2622	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
43		eqid 2622	. . . . . . . . . . 11 |- ( invg ` H ) = ( invg ` H )
44	1, 42, 43	subginv 17601	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
45	21, 41, 44	syl2anc 693	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
46	20, 45	syldan 487	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
47	9	adantr 481	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
48	47	fveq1d 6193	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) )
49	48	fveq2d 6195	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
50	46, 49	eqtrd 2656	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
51	50	anassrs 680	. . . . . 6 |- ( ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
52	51	ifeq2da 4117	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
53	12, 52	eqtrd 2656	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
54	53	ifeq2da 4117	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
55	5, 54	eqtrd 2656	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
56	28, 6, 2, 42, 34, 35	mulgval 17543	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` G ) ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
57	13, 32, 56	syl2anc 693	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
58	1	subgbas 17598	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
59	58	3ad2ant1 1082	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S = ( Base ` H ) )
60	31, 59	eleqtrd 2703	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` H ) )
61		eqid 2622	. . . 4 |- ( Base ` H ) = ( Base ` H )
62		eqid 2622	. . . 4 |- ( +g ` H ) = ( +g ` H )
63		eqid 2622	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
64		subgmulg.t	. . . 4 |- .xb = (.g ` H )
65		eqid 2622	. . . 4 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
66	61, 62, 63, 43, 64, 65	mulgval 17543	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` H ) ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
67	13, 60, 66	syl2anc 693	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
68	55, 57, 67	3eqtr4d 2666	1 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   -ucneg 10267   NNcn 11020   ZZcz 11377   seqcseq 12801   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   invgcminusg 17423  ._gcmg 17540  SubGrpcsubg 17588

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-inf2 8538  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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591

This theorem is referenced by:  cycsubgcyg  18302  ablfac2  18488  zringmulg  19826  zringcyg  19839  remulg  19953  rezh  30015