Theorem mulgnn0subcl 15652

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 15651	. . . . 5 |- ( ( ph /\ N e. NN /\ X e. S ) -> ( N .x. X ) e. S )
8	7	3expa 1187	. . . 4 |- ( ( ( ph /\ N e. NN ) /\ X e. S ) -> ( N .x. X ) e. S )
9	8	an32s 802	. . 3 |- ( ( ( ph /\ X e. S ) /\ N e. NN ) -> ( N .x. X ) e. S )
10	9	3adantl2 1145	. 2 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N e. NN ) -> ( N .x. X ) e. S )
11		oveq1 6110	. . . 4 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
12	5	3ad2ant1 1009	. . . . . 6 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> S C_ B )
13		simp3 990	. . . . . 6 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> X e. S )
14	12, 13	sseldd 3369	. . . . 5 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> X e. B )
15		mulgnn0subcl.z	. . . . . 6 |- .0. = ( 0g ` G )
16	1, 15, 2	mulg0 15644	. . . . 5 |- ( X e. B -> ( 0 .x. X ) = .0. )
17	14, 16	syl 16	. . . 4 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( 0 .x. X ) = .0. )
18	11, 17	sylan9eqr 2497	. . 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 1009	. . . 4 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> .0. e. S )
21	20	adantr 465	. . 3 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> .0. e. S )
22	18, 21	eqeltrd 2517	. 2 |- ( ( ( ph /\ N e. NN0 /\ X e. S ) /\ N = 0 ) -> ( N .x. X ) e. S )
23		simp2 989	. . 3 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> N e. NN0 )
24		elnn0 10593	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
25	23, 24	sylib 196	. 2 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N e. NN \/ N = 0 ) )
26	10, 22, 25	mpjaodan 784	1 |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   C_ wss 3340   ` cfv 5430  (class class class)co 6103   0cc0 9294   NNcn 10334   NN0cn0 10591   Basecbs 14186   +g cplusg 14250   0gc0g 14390  ._gcmg 15426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-seq 11819  df-mulg 15560

This theorem is referenced by:  mulgsubcl  15653  mulgnn0cl  15655  submmulgcl  15673  mplbas2  17563  mplbas2OLD  17564