Theorem mulgsubcl 13786

Description: Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
mulgnnsubcl.b	|- B = ( Base ` G )
mulgnnsubcl.t	|- .x. = (.g ` G )
mulgnnsubcl.p	|- .+ = ( +g ` G )
mulgnnsubcl.g	|- ( ph -> G e. V )
mulgnnsubcl.s	|- ( ph -> S C_ B )
mulgnnsubcl.c	|- ( ( ph /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )
mulgnn0subcl.z	|- .0. = ( 0g ` G )
mulgnn0subcl.c	|- ( ph -> .0. e. S )
mulgsubcl.i	|- I = ( invg ` G )
mulgsubcl.c	|- ( ( ph /\ x e. S ) -> ( I ` x ) e. S )

Assertion

Ref	Expression
mulgsubcl	|- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )

Distinct variable groups:   x, y, .+   x, B, y   x, G, y   x, I   x, N, y   x, S, y   ph, x, y   x, .x.   x, X, y

Allowed substitution hints:   .x. ( y)   I( y)   V( x, y)   .0. ( x, y)

Proof of Theorem mulgsubcl

Step	Hyp	Ref	Expression
1		mulgnnsubcl.b	. . . . . 6 |- B = ( Base ` G )
2		mulgnnsubcl.t	. . . . . 6 |- .x. = (.g ` G )
3		mulgnnsubcl.p	. . . . . 6 |- .+ = ( +g ` G )
4		mulgnnsubcl.g	. . . . . 6 |- ( ph -> G e. V )
5		mulgnnsubcl.s	. . . . . 6 |- ( ph -> S C_ B )
6		mulgnnsubcl.c	. . . . . 6 |- ( ( ph /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )
7		mulgnn0subcl.z	. . . . . 6 |- .0. = ( 0g ` G )
8		mulgnn0subcl.c	. . . . . 6 |- ( ph -> .0. e. S )
9	1, 2, 3, 4, 5, 6, 7, 8	mulgnn0subcl 13785	. . . . 5 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )
10	9	3expa 1230	. . . 4 |- ( ( ( ph /\ N e. NN0 ) /\ X e. S ) -> ( N .x. X ) e. S )
11	10	an32s 570	. . 3 |- ( ( ( ph /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
12	11	3adantl2 1181	. 2 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
13		simp2 1025	. . . . . . . . 9 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> N e. ZZ )
14	13	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> N e. ZZ )
15	14	zcnd 9647	. . . . . . 7 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> N e. CC )
16	15	negnegd 8523	. . . . . 6 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> -u -u N = N )
17	16	oveq1d 6043	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( -u -u N .x. X ) = ( N .x. X ) )
18		id 19	. . . . . 6 |- ( -u N e. NN -> -u N e. NN )
19	5	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> S C_ B )
20		simp3 1026	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. S )
21	19, 20	sseldd 3229	. . . . . 6 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. B )
22		mulgsubcl.i	. . . . . . 7 |- I = ( invg ` G )
23	1, 2, 22	mulgnegnn 13782	. . . . . 6 |- ( ( -u N e. NN /\ X e. B ) -> ( -u -u N .x. X ) = ( I ` ( -u N .x. X ) ) )
24	18, 21, 23	syl2anr 290	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( -u -u N .x. X ) = ( I ` ( -u N .x. X ) ) )
25	17, 24	eqtr3d 2266	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( N .x. X ) = ( I ` ( -u N .x. X ) ) )
26		fveq2 5648	. . . . . 6 |- ( x = ( -u N .x. X ) -> ( I ` x ) = ( I ` ( -u N .x. X ) ) )
27	26	eleq1d 2300	. . . . 5 |- ( x = ( -u N .x. X ) -> ( ( I ` x ) e. S <-> ( I ` ( -u N .x. X ) ) e. S ) )
28		mulgsubcl.c	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> ( I ` x ) e. S )
29	28	ralrimiva 2606	. . . . . . 7 |- ( ph -> A. x e. S ( I ` x ) e. S )
30	29	3ad2ant1 1045	. . . . . 6 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> A. x e. S ( I ` x ) e. S )
31	30	adantr 276	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> A. x e. S ( I ` x ) e. S )
32	1, 2, 3, 4, 5, 6	mulgnnsubcl 13784	. . . . . . . 8 |- ( ( ph /\ -u N e. NN /\ X e. S ) -> ( -u N .x. X ) e. S )
33	32	3expa 1230	. . . . . . 7 |- ( ( ( ph /\ -u N e. NN ) /\ X e. S ) -> ( -u N .x. X ) e. S )
34	33	an32s 570	. . . . . 6 |- ( ( ( ph /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
35	34	3adantl2 1181	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
36	27, 31, 35	rspcdva 2916	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( I ` ( -u N .x. X ) ) e. S )
37	25, 36	eqeltrd 2308	. . 3 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( N .x. X ) e. S )
38	37	adantrl 478	. 2 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( N .x. X ) e. S )
39		elznn0nn 9537	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
40	13, 39	sylib 122	. 2 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
41	12, 38, 40	mpjaodan 806	1 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   ` cfv 5333  (class class class)co 6028   RRcr 8074   -ucneg 8393   NNcn 9185   NN0cn0 9444   ZZcz 9523   Basecbs 13145   +g cplusg 13223   0gc0g 13402   invgcminusg 13647  ._gcmg 13769

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-uz 9800  df-seqfrec 10756  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-0g 13404  df-minusg 13650  df-mulg 13770

This theorem is referenced by:  mulgcl  13789  subgmulgcl  13837