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Theorem mulgnn0subcl 14832
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
StepHypRef 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
)
71, 2, 3, 4, 5, 6mulgnnsubcl 14831 . . . . 5  |-  ( (
ph  /\  N  e.  NN  /\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
873expa 1153 . . . 4  |-  ( ( ( ph  /\  N  e.  NN )  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
98an32s 780 . . 3  |-  ( ( ( ph  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .x.  X )  e.  S )
1093adantl2 1114 . 2  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .x.  X )  e.  S )
11 oveq1 6029 . . . 4  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
1253ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  S  C_  B
)
13 simp3 959 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  X  e.  S )
1412, 13sseldd 3294 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  X  e.  B )
15 mulgnn0subcl.z . . . . . 6  |-  .0.  =  ( 0g `  G )
161, 15, 2mulg0 14824 . . . . 5  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
1714, 16syl 16 . . . 4  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( 0 
.x.  X )  =  .0.  )
1811, 17sylan9eqr 2443 . . 3  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  .0.  )
19 mulgnn0subcl.c . . . . 5  |-  ( ph  ->  .0.  e.  S )
20193ad2ant1 978 . . . 4  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  .0.  e.  S )
2120adantr 452 . . 3  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  .0.  e.  S )
2218, 21eqeltrd 2463 . 2  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  e.  S )
23 simp2 958 . . 3  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  N  e.  NN0 )
24 elnn0 10157 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2523, 24sylib 189 . 2  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  e.  NN  \/  N  =  0 ) )
2610, 22, 25mpjaodan 762 1  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396  (class class class)co 6022   0cc0 8925   NNcn 9934   NN0cn0 10155   Basecbs 13398   +g cplusg 13458   0gc0g 13652  .gcmg 14618
This theorem is referenced by:  mulgsubcl  14833  mulgnn0cl  14835  submmulgcl  14853  mplbas2  16460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-seq 11253  df-mulg 14744
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