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Theorem mulgsubcl 14581
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 14580 . . . . 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 10118 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  CC )
1615negnegd 9148 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  -u -u N  =  N )
1716oveq1d 5873 . . . . 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 3181 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  B )
22 mulgsubcl.i . . . . . . 7  |-  I  =  ( inv g `  G )
231, 2, 22mulgnegnn 14577 . . . . . 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 2317 . . . 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 14579 . . . . . . . 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 2626 . . . . . . 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 5525 . . . . . . 7  |-  ( x  =  ( -u N  .x.  X )  ->  (
I `  x )  =  ( I `  ( -u N  .x.  X
) ) )
3534eleq1d 2349 . . . . . 6  |-  ( x  =  ( -u N  .x.  X )  ->  (
( I `  x
)  e.  S  <->  ( I `  ( -u N  .x.  X ) )  e.  S ) )
3635rspcv 2880 . . . . 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 2357 . . 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 10037 . . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   RRcr 8736   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   Basecbs 13148   +g cplusg 13208   0gc0g 13400   inv gcminusg 14363  .gcmg 14366
This theorem is referenced by:  mulgcl  14584  subgmulgcl  14634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-mulg 14492
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