Theorem mulgsubcl 13668

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 13667	. . . . 5 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )
10	9	3expa 1227	. . . 4 |- ( ( ( ph /\ N e. NN0 ) /\ X e. S ) -> ( N .x. X ) e. S )
11	10	an32s 568	. . 3 |- ( ( ( ph /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
12	11	3adantl2 1178	. 2 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ N e. NN0 ) -> ( N .x. X ) e. S )
13		simp2 1022	. . . . . . . . 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 9566	. . . . . . 7 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> N e. CC )
16	15	negnegd 8444	. . . . . 6 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> -u -u N = N )
17	16	oveq1d 6015	. . . . 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 1042	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> S C_ B )
20		simp3 1023	. . . . . . 7 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. S )
21	19, 20	sseldd 3225	. . . . . 6 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> X e. B )
22		mulgsubcl.i	. . . . . . 7 |- I = ( invg ` G )
23	1, 2, 22	mulgnegnn 13664	. . . . . 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 2264	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( N .x. X ) = ( I ` ( -u N .x. X ) ) )
26		fveq2 5626	. . . . . 6 |- ( x = ( -u N .x. X ) -> ( I ` x ) = ( I ` ( -u N .x. X ) ) )
27	26	eleq1d 2298	. . . . 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 2603	. . . . . . 7 |- ( ph -> A. x e. S ( I ` x ) e. S )
30	29	3ad2ant1 1042	. . . . . 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 13666	. . . . . . . 8 |- ( ( ph /\ -u N e. NN /\ X e. S ) -> ( -u N .x. X ) e. S )
33	32	3expa 1227	. . . . . . 7 |- ( ( ( ph /\ -u N e. NN ) /\ X e. S ) -> ( -u N .x. X ) e. S )
34	33	an32s 568	. . . . . 6 |- ( ( ( ph /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
35	34	3adantl2 1178	. . . . 5 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( -u N .x. X ) e. S )
36	27, 31, 35	rspcdva 2912	. . . 4 |- ( ( ( ph /\ N e. ZZ /\ X e. S ) /\ -u N e. NN ) -> ( I ` ( -u N .x. X ) ) e. S )
37	25, 36	eqeltrd 2306	. . 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 9456	. . 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 803	1 |- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   ` cfv 5317  (class class class)co 6000   RRcr 7994   -ucneg 8314   NNcn 9106   NN0cn0 9365   ZZcz 9442   Basecbs 13027   +g cplusg 13105   0gc0g 13284   invgcminusg 13529  ._gcmg 13651

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-minusg 13532  df-mulg 13652

This theorem is referenced by:  mulgcl  13671  subgmulgcl  13719