Theorem submmulg 13296

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 2197	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S e. (SubMnd ` G ) )
6		submrcl 13103	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
7	3, 4, 5, 6	ressplusgd 12806	. . . . . 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 10546	. . . 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 5560	. . 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 2196	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
13	12	submss 13108	. . . . . . 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 3184	. . . . 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 2196	. . . . 5 |- ( +g ` G ) = ( +g ` G )
19		submmulgcl.t	. . . . 5 |- .xb = (.g ` G )
20		eqid 2196	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21	12, 18, 19, 20	mulgnn 13256	. . . 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 13113	. . . . . . 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 2275	. . . . 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 2196	. . . . 5 |- ( Base ` H ) = ( Base ` H )
28		eqid 2196	. . . . 5 |- ( +g ` H ) = ( +g ` H )
29		submmulg.t	. . . . 5 |- .x. = (.g ` H )
30		eqid 2196	. . . . 5 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
31	27, 28, 29, 30	mulgnn 13256	. . . 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 2239	. 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 2196	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
36	2, 35	subm0 13114	. . . . 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 13255	. . . . 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 2196	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
43	27, 42, 29	mulg0 13255	. . . . 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 2239	. . 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 5937	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( 0 .xb X ) )
48	46	oveq1d 5937	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
49	45, 47, 48	3eqtr4d 2239	. 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 9251	. . 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 2167   C_ wss 3157   {csn 3622   X. cxp 4661   ` cfv 5258  (class class class)co 5922   0cc0 7879   1c1 7880   NNcn 8990   NN0cn0 9249   seqcseq 10539   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057  SubMndcsubmnd 13090  ._gcmg 13249

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-submnd 13092  df-minusg 13136  df-mulg 13250

This theorem is referenced by:  lgseisenlem4  15314