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Theorem mulgsubcl 16358
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  =  ( invg `  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 16357 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
1093expa 1194 . . . 4  |-  ( ( ( ph  /\  N  e.  NN0 )  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
1110an32s 802 . . 3  |-  ( ( ( ph  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
12113adantl2 1151 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
13 simp2 995 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  N  e.  ZZ )
1413adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  ZZ )
1514zcnd 10966 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  CC )
1615negnegd 9913 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  -u -u N  =  N )
1716oveq1d 6285 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u -u N  .x.  X )  =  ( N  .x.  X ) )
18 id 22 . . . . . 6  |-  ( -u N  e.  NN  ->  -u N  e.  NN )
1953ad2ant1 1015 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  S  C_  B
)
20 simp3 996 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  S )
2119, 20sseldd 3490 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  B )
22 mulgsubcl.i . . . . . . 7  |-  I  =  ( invg `  G )
231, 2, 22mulgnegnn 16354 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
2418, 21, 23syl2anr 476 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X
) ) )
2517, 24eqtr3d 2497 . . . 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 16356 . . . . . . . 8  |-  ( (
ph  /\  -u N  e.  NN  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
27263expa 1194 . . . . . . 7  |-  ( ( ( ph  /\  -u N  e.  NN )  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
2827an32s 802 . . . . . 6  |-  ( ( ( ph  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u N  .x.  X )  e.  S )
29283adantl2 1151 . . . . 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 2868 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( I `  x
)  e.  S )
32313ad2ant1 1015 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  A. x  e.  S  ( I `  x )  e.  S
)
3332adantr 463 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  A. x  e.  S  ( I `  x )  e.  S
)
34 fveq2 5848 . . . . . . 7  |-  ( x  =  ( -u N  .x.  X )  ->  (
I `  x )  =  ( I `  ( -u N  .x.  X
) ) )
3534eleq1d 2523 . . . . . 6  |-  ( x  =  ( -u N  .x.  X )  ->  (
( I `  x
)  e.  S  <->  ( I `  ( -u N  .x.  X ) )  e.  S ) )
3635rspcv 3203 . . . . 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 60 . . . 4  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  (
I `  ( -u N  .x.  X ) )  e.  S )
3825, 37eqeltrd 2542 . . 3  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( N  .x.  X )  e.  S )
3938adantrl 713 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( N  .x.  X
)  e.  S )
40 elznn0nn 10874 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4113, 40sylib 196 . 2  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4212, 39, 41mpjaodan 784 1  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   ` cfv 5570  (class class class)co 6270   RRcr 9480   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860   Basecbs 14719   +g cplusg 14787   0gc0g 14932   invgcminusg 16256  .gcmg 16258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12093  df-mulg 16262
This theorem is referenced by:  mulgcl  16361  subgmulgcl  16416
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