MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgsubcl Structured version   Unicode version

Theorem mulgsubcl 14906
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 14905 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 
/\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
1093expa 1154 . . . 4  |-  ( ( ( ph  /\  N  e.  NN0 )  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
1110an32s 781 . . 3  |-  ( ( ( ph  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
12113adantl2 1115 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  e.  NN0 )  ->  ( N  .x.  X )  e.  S )
13 simp2 959 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  N  e.  ZZ )
1413adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  ZZ )
1514zcnd 10378 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  N  e.  CC )
1615negnegd 9404 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  -u -u N  =  N )
1716oveq1d 6098 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u -u N  .x.  X )  =  ( N  .x.  X ) )
18 id 21 . . . . . 6  |-  ( -u N  e.  NN  ->  -u N  e.  NN )
1953ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  S  C_  B
)
20 simp3 960 . . . . . . 7  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  S )
2119, 20sseldd 3351 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  X  e.  B )
22 mulgsubcl.i . . . . . . 7  |-  I  =  ( inv g `  G )
231, 2, 22mulgnegnn 14902 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
2418, 21, 23syl2anr 466 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X
) ) )
2517, 24eqtr3d 2472 . . . 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 14904 . . . . . . . 8  |-  ( (
ph  /\  -u N  e.  NN  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
27263expa 1154 . . . . . . 7  |-  ( ( ( ph  /\  -u N  e.  NN )  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
2827an32s 781 . . . . . 6  |-  ( ( ( ph  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( -u N  .x.  X )  e.  S )
29283adantl2 1115 . . . . 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 2791 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( I `  x
)  e.  S )
32313ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  A. x  e.  S  ( I `  x )  e.  S
)
3332adantr 453 . . . . 5  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  A. x  e.  S  ( I `  x )  e.  S
)
34 fveq2 5730 . . . . . . 7  |-  ( x  =  ( -u N  .x.  X )  ->  (
I `  x )  =  ( I `  ( -u N  .x.  X
) ) )
3534eleq1d 2504 . . . . . 6  |-  ( x  =  ( -u N  .x.  X )  ->  (
( I `  x
)  e.  S  <->  ( I `  ( -u N  .x.  X ) )  e.  S ) )
3635rspcv 3050 . . . . 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 59 . . . 4  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  (
I `  ( -u N  .x.  X ) )  e.  S )
3825, 37eqeltrd 2512 . . 3  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  -u N  e.  NN )  ->  ( N  .x.  X )  e.  S )
3938adantrl 698 . 2  |-  ( ( ( ph  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( N  .x.  X
)  e.  S )
40 elznn0nn 10297 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4113, 40sylib 190 . 2  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4212, 39, 41mpjaodan 763 1  |-  ( (
ph  /\  N  e.  ZZ  /\  X  e.  S
)  ->  ( N  .x.  X )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ` cfv 5456  (class class class)co 6083   RRcr 8991   -ucneg 9294   NNcn 10002   NN0cn0 10223   ZZcz 10284   Basecbs 13471   +g cplusg 13531   0gc0g 13725   inv gcminusg 14688  .gcmg 14691
This theorem is referenced by:  mulgcl  14909  subgmulgcl  14959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-seq 11326  df-mulg 14817
  Copyright terms: Public domain W3C validator