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Theorem mulgsubcl 14597
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  =  ( inv g `  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
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
)
7 mulgnn0subcl.z . . . . . 6  |-  .0.  =  ( 0g `  G )
8 mulgnn0subcl.c . . . . . 6  |-  ( ph  ->  .0.  e.  S )
91, 2, 3, 4, 5, 6, 7, 8mulgnn0subcl 14596 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
1093expa 1151 . . . 4  |-  ( ( ( ph  /\  N  e.  NN0 )  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
1110an32s 779 . . 3  |-  ( ( ( ph  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
12113adantl2 1112 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
13 simp2 956 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  N  e.  ZZ )
1413adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  ZZ )
1514zcnd 10134 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  CC )
1615negnegd 9164 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  -u -u N  =  N )
1716oveq1d 5889 . . . . 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 )
1953ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  S  C_  B
)
20 simp3 957 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  S )
2119, 20sseldd 3194 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  B )
22 mulgsubcl.i . . . . . . 7  |-  I  =  ( inv g `  G )
231, 2, 22mulgnegnn 14593 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
2418, 21, 23syl2anr 464 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X
) ) )
2517, 24eqtr3d 2330 . . . 4  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
261, 2, 3, 4, 5, 6mulgnnsubcl 14595 . . . . . . . 8  |-  ( (
ph  /\  -u N  e.  NN  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
27263expa 1151 . . . . . . 7  |-  ( ( ( ph  /\  -u N  e.  NN )  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
2827an32s 779 . . . . . 6  |-  ( ( ( ph  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u N  .x.  X )  e.  S )
29283adantl2 1112 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u N  .x.  X )  e.  S )
30 mulgsubcl.c . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  (
I `  x )  e.  S )
3130ralrimiva 2639 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( I `  x
)  e.  S )
32313ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  A. x  e.  S  ( I `  x )  e.  S
)
3332adantr 451 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  A. x  e.  S  ( I `  x )  e.  S
)
34 fveq2 5541 . . . . . . 7  |-  ( x  =  ( -u N  .x.  X )  ->  (
I `  x )  =  ( I `  ( -u N  .x.  X
) ) )
3534eleq1d 2362 . . . . . 6  |-  ( x  =  ( -u N  .x.  X )  ->  (
( I `  x
)  e.  S  <->  ( I `  ( -u N  .x.  X ) )  e.  S ) )
3635rspcv 2893 . . . . 5  |-  ( (
-u N  .x.  X
)  e.  S  -> 
( A. x  e.  S  ( I `  x )  e.  S  ->  ( I `  ( -u N  .x.  X ) )  e.  S ) )
3729, 33, 36sylc 56 . . . 4  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  (
I `  ( -u N  .x.  X ) )  e.  S )
3825, 37eqeltrd 2370 . . 3  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( N  .x.  X )  e.  S )
3938adantrl 696 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( N  .x.  X
)  e.  S )
40 elznn0nn 10053 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4113, 40sylib 188 . 2  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4212, 39, 41mpjaodan 761 1  |-  ( (
ph  /\  N  e.  ZZ  /\  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   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   RRcr 8752   -ucneg 9054   NNcn 9762   NN0cn0 9981   ZZcz 10040   Basecbs 13164   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379  .gcmg 14382
This theorem is referenced by:  mulgcl  14600  subgmulgcl  14650
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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