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Theorem mulgnn0subcl 14596
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 14595 . . . . 5  |-  ( (
ph  /\  N  e.  NN  /\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
873expa 1151 . . . 4  |-  ( ( ( ph  /\  N  e.  NN )  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
98an32s 779 . . 3  |-  ( ( ( ph  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .x.  X )  e.  S )
1093adantl2 1112 . 2  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  e.  NN )  ->  ( N  .x.  X )  e.  S )
11 oveq1 5881 . . . 4  |-  ( N  =  0  ->  ( N  .x.  X )  =  ( 0  .x.  X
) )
1253ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  S  C_  B
)
13 simp3 957 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  X  e.  S )
1412, 13sseldd 3194 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  X  e.  B )
15 mulgnn0subcl.z . . . . . 6  |-  .0.  =  ( 0g `  G )
161, 15, 2mulg0 14588 . . . . 5  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
1714, 16syl 15 . . . 4  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( 0 
.x.  X )  =  .0.  )
1811, 17sylan9eqr 2350 . . 3  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  =  .0.  )
19 mulgnn0subcl.c . . . . 5  |-  ( ph  ->  .0.  e.  S )
20193ad2ant1 976 . . . 4  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  .0.  e.  S )
2120adantr 451 . . 3  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  ->  .0.  e.  S )
2218, 21eqeltrd 2370 . 2  |-  ( ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  /\  N  =  0 )  -> 
( N  .x.  X
)  e.  S )
23 simp2 956 . . 3  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  N  e.  NN0 )
24 elnn0 9983 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2523, 24sylib 188 . 2  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  e.  NN  \/  N  =  0 ) )
2610, 22, 25mpjaodan 761 1  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   0cc0 8753   NNcn 9762   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   0gc0g 13416  .gcmg 14382
This theorem is referenced by:  mulgsubcl  14597  mulgnn0cl  14599  submmulgcl  14617  mplbas2  16228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-mulg 14508
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