Theorem submmulg 13546

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 1003	. . . . . 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 2207	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S e. (SubMnd ` G ) )
6		submrcl 13347	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> G e. Mnd )
7	3, 4, 5, 6	ressplusgd 13005	. . . . . 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 10606	. . . 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 5585	. . 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 2206	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
13	12	submss 13352	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
14	13	3ad2ant1 1021	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S C_ ( Base ` G ) )
15		simp3 1002	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> X e. S )
16	14, 15	sseldd 3195	. . . . 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 2206	. . . . 5 |- ( +g ` G ) = ( +g ` G )
19		submmulgcl.t	. . . . 5 |- .xb = (.g ` G )
20		eqid 2206	. . . . 5 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21	12, 18, 19, 20	mulgnn 13506	. . . 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 13357	. . . . . . 7 |- ( S e. (SubMnd ` G ) -> S = ( Base ` H ) )
24	23	3ad2ant1 1021	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> S = ( Base ` H ) )
25	15, 24	eleqtrd 2285	. . . . 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 2206	. . . . 5 |- ( Base ` H ) = ( Base ` H )
28		eqid 2206	. . . . 5 |- ( +g ` H ) = ( +g ` H )
29		submmulg.t	. . . . 5 |- .x. = (.g ` H )
30		eqid 2206	. . . . 5 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
31	27, 28, 29, 30	mulgnn 13506	. . . 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 2249	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .xb X ) = ( N .x. X ) )
34		simpl1 1003	. . . . 5 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> S e. (SubMnd ` G ) )
35		eqid 2206	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
36	2, 35	subm0 13358	. . . . 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 13505	. . . . 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 2206	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
43	27, 42, 29	mulg0 13505	. . . . 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 2249	. . 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 5966	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( 0 .xb X ) )
48	46	oveq1d 5966	. . 3 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) )
49	45, 47, 48	3eqtr4d 2249	. 2 |- ( ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .xb X ) = ( N .x. X ) )
50		simp2 1001	. . 3 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
51		elnn0 9304	. . 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 800	1 |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2177   C_ wss 3167   {csn 3634   X. cxp 4677   ` cfv 5276  (class class class)co 5951   0cc0 7932   1c1 7933   NNcn 9043   NN0cn0 9302   seqcseq 10599   Basecbs 12876   ↾_s cress 12877   +g cplusg 12953   0gc0g 13132   Mndcmnd 13292  SubMndcsubmnd 13334  ._gcmg 13499

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-2 9102  df-n0 9303  df-z 9380  df-uz 9656  df-seqfrec 10600  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-submnd 13336  df-minusg 13380  df-mulg 13500

This theorem is referenced by:  lgseisenlem4  15594