Theorem mulgnn0subcl 13802

Description: Closure of the group multiple (exponentiation) operation in a submonoid. (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 )

Assertion

Ref	Expression
mulgnn0subcl	|- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )

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

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

Proof of Theorem mulgnn0subcl

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	1, 2, 3, 4, 5, 6	mulgnnsubcl 13801	. . . . 5 |- ( ( ph /\ N e. NN /\ X e. S ) -> ( N .x. X ) e. S )
8	7	3expa 1230	. . . 4 |- ( ( ( ph /\ N e. NN ) /\ X e. S ) -> ( N .x. X ) e. S )
9	8	an32s 570	. . 3 |- ( ( ( ph /\ X e. S ) /\ N e. NN ) -> ( N .x. X ) e. S )
10	9	3adantl2 1181	. 2 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .x. X ) e. S )
11		oveq1 6035	. . . 4 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
12	5	3ad2ant1 1045	. . . . . 6 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> S C_ B )
13		simp3 1026	. . . . . 6 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> X e. S )
14	12, 13	sseldd 3229	. . . . 5 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> X e. B )
15		mulgnn0subcl.z	. . . . . 6 |- .0. = ( 0g ` G )
16	1, 15, 2	mulg0 13792	. . . . 5 |- ( X e. B -> ( 0 .x. X ) = .0. )
17	14, 16	syl 14	. . . 4 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( 0 .x. X ) = .0. )
18	11, 17	sylan9eqr 2286	. . 3 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) = .0. )
19		mulgnn0subcl.c	. . . . 5 |- ( ph -> .0. e. S )
20	19	3ad2ant1 1045	. . . 4 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> .0. e. S )
21	20	adantr 276	. . 3 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> .0. e. S )
22	18, 21	eqeltrd 2308	. 2 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) e. S )
23		simp2 1025	. . 3 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
24		elnn0 9463	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
25	23, 24	sylib 122	. 2 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N e. NN \/ N = 0 ) )
26	10, 22, 25	mpjaodan 806	1 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   ` cfv 5333  (class class class)co 6028   0cc0 8092   NNcn 9202   NN0cn0 9461   Basecbs 13162   +g cplusg 13240   0gc0g 13419  ._gcmg 13786

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 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

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 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-minusg 13667  df-mulg 13787

This theorem is referenced by:  mulgsubcl  13803  mulgnn0cl  13805  submmulgcl  13832