Theorem submmulg 13969

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 1031	. . . . . 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 2239	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S e. (SubMnd ` G ) )
6		submrcl 13778	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
7	3, 4, 5, 6	ressplusgd 13483	. . . . . 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 10891	. . . 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 5697	. . 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 2238	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
13	12	submss 13783	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
14	13	3ad2ant1 1049	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S C_ ( Base ` G ) )
15		simp3 1030	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. S )
16	14, 15	sseldd 3249	. . . . 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 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
19		submmulgcl.t	. . . . 5 |- .xb = (.g ` G )
20		eqid 2238	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21	12, 18, 19, 20	mulgnn 13929	. . . 4 |- ( ( N e. NN /\ X e. ( Base ` G ) ) -> ( N .xb X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
22	11, 17, 21	syl2anc 415	. . 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 13788	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S = ( Base ` H ) )
24	23	3ad2ant1 1049	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S = ( Base ` H ) )
25	15, 24	eleqtrd 2317	. . . . 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 2238	. . . . 5 |- ( Base ` H ) = ( Base ` H )
28		eqid 2238	. . . . 5 |- ( +g ` H ) = ( +g ` H )
29		submmulg.t	. . . . 5 |- .x. = (.g ` H )
30		eqid 2238	. . . . 5 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
31	27, 28, 29, 30	mulgnn 13929	. . . 4 |- ( ( N e. NN /\ X e. ( Base ` H ) ) -> ( N .x. X ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
32	11, 26, 31	syl2anc 415	. . 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 2281	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .xb X ) = ( N .x. X ) )
34		simpl1 1031	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> S e. (SubMnd ` G ) )
35		eqid 2238	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
36	2, 35	subm0 13789	. . . . 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 13928	. . . . 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 2238	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
43	27, 42, 29	mulg0 13928	. . . . 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 2281	. . 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 6100	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( 0 .xb X ) )
48	46	oveq1d 6100	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
49	45, 47, 48	3eqtr4d 2281	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( N .x. X ) )
50		simp2 1029	. . 3 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
51		elnn0 9565	. . 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 810	1 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   {csn 3709   X. cxp 4772   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   NNcn 9304   NN0cn0 9563   seqcseq 10884   Basecbs 13352   ↾_s cress 13353   +g cplusg 13431   0gc0g 13610   Mndcmnd 13729  SubMndcsubmnd 13765  ._gcmg 13922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-submnd 13767  df-minusg 13809  df-mulg 13923

This theorem is used by:  lgseisenlem4  16192