Theorem mulgsubcl 13916

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 13915	. . . . 5 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )
10	9	3expa 1234	. . . 4 |- ( ( ( ph /\ N e. NN0 ) /\ X e. S ) -> ( N .x. X ) e. S )
11	10	an32s 574	. . 3 |- ( ( ( ph /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
12	11	3adantl2 1185	. 2 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
13		simp2 1029	. . . . . . . . 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 9748	. . . . . . 7 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> N e. CC )
16	15	negnegd 8618	. . . . . 6 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> -u -u N = N )
17	16	oveq1d 6090	. . . . 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 1049	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> S C_ B )
20		simp3 1030	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. S )
21	19, 20	sseldd 3249	. . . . . 6 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. B )
22		mulgsubcl.i	. . . . . . 7 |- I = ( invg ` G )
23	1, 2, 22	mulgnegnn 13912	. . . . . 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 2273	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( N .x. X ) = ( I ` ( -u N .x. X ) ) )
26		fveq2 5690	. . . . . 6 |- ( x = ( -u N .x. X ) -> ( I ` x ) = ( I ` ( -u N .x. X ) ) )
27	26	eleq1d 2307	. . . . 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 2623	. . . . . . 7 |- ( ph -> A. x e. S ( I ` x ) e. S )
30	29	3ad2ant1 1049	. . . . . 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 13914	. . . . . . . 8 |- ( ( ph /\ -u N e. NN /\ X e. S ) -> ( -u N .x. X ) e. S )
33	32	3expa 1234	. . . . . . 7 |- ( ( ( ph /\ -u N e. NN ) /\ X e. S ) -> ( -u N .x. X ) e. S )
34	33	an32s 574	. . . . . 6 |- ( ( ( ph /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
35	34	3adantl2 1185	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
36	27, 31, 35	rspcdva 2934	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( I ` ( -u N .x. X ) ) e. S )
37	25, 36	eqeltrd 2315	. . 3 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( N .x. X ) e. S )
38	37	adantrl 482	. 2 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( N .x. X ) e. S )
39		elznn0nn 9637	. . 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 810	1 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5372  (class class class)co 6075   RRcr 8168   -ucneg 8488   NNcn 9283   NN0cn0 9542   ZZcz 9623   Basecbs 13330   +g cplusg 13408   0gc0g 13587   invgcminusg 13783  ._gcmg 13899

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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-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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-minusg 13786  df-mulg 13900

This theorem is referenced by:  mulgcl  13919  subgmulgcl  13967