Theorem submmulg 13883

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 1027	. . . . . 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 2233	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S e. (SubMnd ` G ) )
6		submrcl 13684	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
7	3, 4, 5, 6	ressplusgd 13342	. . . . . 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 10816	. . . 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 5672	. . 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 2232	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
13	12	submss 13689	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
14	13	3ad2ant1 1045	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S C_ ( Base ` G ) )
15		simp3 1026	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. S )
16	14, 15	sseldd 3239	. . . . 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 2232	. . . . 5 |- ( +g ` G ) = ( +g ` G )
19		submmulgcl.t	. . . . 5 |- .xb = (.g ` G )
20		eqid 2232	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21	12, 18, 19, 20	mulgnn 13843	. . . 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 13694	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S = ( Base ` H ) )
24	23	3ad2ant1 1045	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S = ( Base ` H ) )
25	15, 24	eleqtrd 2311	. . . . 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 2232	. . . . 5 |- ( Base ` H ) = ( Base ` H )
28		eqid 2232	. . . . 5 |- ( +g ` H ) = ( +g ` H )
29		submmulg.t	. . . . 5 |- .x. = (.g ` H )
30		eqid 2232	. . . . 5 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
31	27, 28, 29, 30	mulgnn 13843	. . . 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 2275	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .xb X ) = ( N .x. X ) )
34		simpl1 1027	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> S e. (SubMnd ` G ) )
35		eqid 2232	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
36	2, 35	subm0 13695	. . . . 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 13842	. . . . 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 2232	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
43	27, 42, 29	mulg0 13842	. . . . 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 2275	. . 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 6065	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( 0 .xb X ) )
48	46	oveq1d 6065	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
49	45, 47, 48	3eqtr4d 2275	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( N .x. X ) )
50		simp2 1025	. . 3 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
51		elnn0 9498	. . 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 806	1 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   {csn 3689   X. cxp 4747   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   NNcn 9237   NN0cn0 9496   seqcseq 10809   Basecbs 13212   ↾_s cress 13213   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629  SubMndcsubmnd 13671  ._gcmg 13836

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-submnd 13673  df-minusg 13717  df-mulg 13837

This theorem is referenced by:  lgseisenlem4  15946