Theorem submmulg 13239

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

Hypotheses

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

Assertion

Ref	Expression
submmulg	|- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )

Proof of Theorem submmulg

Step	Hyp	Ref	Expression
1		simpl1 1002	. . . . . 6 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> S e. (SubMnd ` G ) )
2		submmulg.h	. . . . . . . 8 |- H = ( G ↾s S )
3	2	a1i 9	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> H = ( G ↾s S ) )
4		eqidd 2194	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S e. (SubMnd ` G ) )
6		submrcl 13046	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
7	3, 4, 5, 6	ressplusgd 12749	. . . . . 6 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` H ) )
8	1, 7	syl 14	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( +g ` G ) = ( +g ` H ) )
9	8	seqeq2d 10528	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
10	9	fveq1d 5557	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
11		simpr 110	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> N e. NN )
12		eqid 2193	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
13	12	submss 13051	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
14	13	3ad2ant1 1020	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S C_ ( Base ` G ) )
15		simp3 1001	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. S )
16	14, 15	sseldd 3181	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. ( Base ` G ) )
17	16	adantr 276	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> X e. ( Base ` G ) )
18		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
19		submmulgcl.t	. . . . 5 |- .xb = (.g ` G )
20		eqid 2193	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21	12, 18, 19, 20	mulgnn 13199	. . . 4 |- ( ( N e. NN /\ X e. ( Base ` G ) ) -> ( N .xb X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
22	11, 17, 21	syl2anc 411	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .xb X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
23	2	submbas 13056	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S = ( Base ` H ) )
24	23	3ad2ant1 1020	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S = ( Base ` H ) )
25	15, 24	eleqtrd 2272	. . . . 5 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. ( Base ` H ) )
26	25	adantr 276	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> X e. ( Base ` H ) )
27		eqid 2193	. . . . 5 |- ( Base ` H ) = ( Base ` H )
28		eqid 2193	. . . . 5 |- ( +g ` H ) = ( +g ` H )
29		submmulg.t	. . . . 5 |- .x. = (.g ` H )
30		eqid 2193	. . . . 5 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
31	27, 28, 29, 30	mulgnn 13199	. . . 4 |- ( ( N e. NN /\ X e. ( Base ` H ) ) -> ( N .x. X ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
32	11, 26, 31	syl2anc 411	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .x. X ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
33	10, 22, 32	3eqtr4d 2236	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .xb X ) = ( N .x. X ) )
34		simpl1 1002	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> S e. (SubMnd ` G ) )
35		eqid 2193	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
36	2, 35	subm0 13057	. . . . 5 |- ( S e. (SubMnd ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
37	34, 36	syl 14	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( 0g ` G ) = ( 0g ` H ) )
38	16	adantr 276	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> X e. ( Base ` G ) )
39	12, 35, 19	mulg0 13198	. . . . 5 |- ( X e. ( Base ` G ) -> ( 0 .xb X ) = ( 0g ` G ) )
40	38, 39	syl 14	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( 0 .xb X ) = ( 0g ` G ) )
41	25	adantr 276	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> X e. ( Base ` H ) )
42		eqid 2193	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
43	27, 42, 29	mulg0 13198	. . . . 5 |- ( X e. ( Base ` H ) -> ( 0 .x. X ) = ( 0g ` H ) )
44	41, 43	syl 14	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( 0 .x. X ) = ( 0g ` H ) )
45	37, 40, 44	3eqtr4d 2236	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( 0 .xb X ) = ( 0 .x. X ) )
46		simpr 110	. . . 4 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> N = 0 )
47	46	oveq1d 5934	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( 0 .xb X ) )
48	46	oveq1d 5934	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
49	45, 47, 48	3eqtr4d 2236	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( N .x. X ) )
50		simp2 1000	. . 3 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
51		elnn0 9245	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
52	50, 51	sylib 122	. 2 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N e. NN \/ N = 0 ) )
53	33, 49, 52	mpjaodan 799	1 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3154   {csn 3619   X. cxp 4658   ` cfv 5255  (class class class)co 5919   0cc0 7874   1c1 7875   NNcn 8984   NN0cn0 9243   seqcseq 10521   Basecbs 12621   ↾_s cress 12622   +g cplusg 12698   0gc0g 12870   Mndcmnd 13000  SubMndcsubmnd 13033  ._gcmg 13192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-submnd 13035  df-minusg 13079  df-mulg 13193

This theorem is referenced by:  lgseisenlem4  15230