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Theorem expcl2lem 11320
Description: Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
expcllem.1  |-  F  C_  CC
expcllem.2  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
expcllem.3  |-  1  e.  F
expcl2lem.4  |-  ( ( x  e.  F  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  F )
Assertion
Ref Expression
expcl2lem  |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Distinct variable groups:    x, y, A    x, B    x, F, y
Allowed substitution hint:    B( y)

Proof of Theorem expcl2lem
StepHypRef Expression
1 elznn0nn 10227 . . 3  |-  ( B  e.  ZZ  <->  ( B  e.  NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) ) )
2 expcllem.1 . . . . . . 7  |-  F  C_  CC
3 expcllem.2 . . . . . . 7  |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y
)  e.  F )
4 expcllem.3 . . . . . . 7  |-  1  e.  F
52, 3, 4expcllem 11319 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 424 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 452 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  NN0  ->  ( A ^ B
)  e.  F ) )
8 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 3289 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simprl 733 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1110recnd 9047 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
12 nnnn0 10160 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1312ad2antll 710 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
14 expneg2 11317 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
159, 11, 13, 14syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  =  ( 1  /  ( A ^ -u B ) ) )
16 difss 3417 . . . . . . . 8  |-  ( F 
\  { 0 } )  C_  F
17 simpl 444 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A  e.  F  /\  A  =/=  0
) )
18 eldifsn 3870 . . . . . . . . . 10  |-  ( A  e.  ( F  \  { 0 } )  <-> 
( A  e.  F  /\  A  =/=  0
) )
1917, 18sylibr 204 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  ( F  \  { 0 } ) )
2016, 2sstri 3300 . . . . . . . . . 10  |-  ( F 
\  { 0 } )  C_  CC
2116sseli 3287 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  x  e.  F
)
2216sseli 3287 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  y  e.  F
)
2321, 22, 3syl2an 464 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  F
)
24 eldifsn 3870 . . . . . . . . . . . . 13  |-  ( x  e.  ( F  \  { 0 } )  <-> 
( x  e.  F  /\  x  =/=  0
) )
252sseli 3287 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2625anim1i 552 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x  =/=  0 )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
2724, 26sylbi 188 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
28 eldifsn 3870 . . . . . . . . . . . . 13  |-  ( y  e.  ( F  \  { 0 } )  <-> 
( y  e.  F  /\  y  =/=  0
) )
292sseli 3287 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3029anim1i 552 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y  =/=  0 )  -> 
( y  e.  CC  /\  y  =/=  0 ) )
3128, 30sylbi 188 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  ( y  e.  CC  /\  y  =/=  0 ) )
32 mulne0 9596 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
3327, 31, 32syl2an 464 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  =/=  0
)
34 eldifsn 3870 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  ( F  \  { 0 } )  <-> 
( ( x  x.  y )  e.  F  /\  ( x  x.  y
)  =/=  0 ) )
3523, 33, 34sylanbrc 646 . . . . . . . . . 10  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( F  \  { 0 } ) )
36 ax-1ne0 8992 . . . . . . . . . . 11  |-  1  =/=  0
37 eldifsn 3870 . . . . . . . . . . 11  |-  ( 1  e.  ( F  \  { 0 } )  <-> 
( 1  e.  F  /\  1  =/=  0
) )
384, 36, 37mpbir2an 887 . . . . . . . . . 10  |-  1  e.  ( F  \  {
0 } )
3920, 35, 38expcllem 11319 . . . . . . . . 9  |-  ( ( A  e.  ( F 
\  { 0 } )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  ( F  \  { 0 } ) )
4019, 13, 39syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  ( F 
\  { 0 } ) )
4116, 40sseldi 3289 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  F )
42 eldifsn 3870 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  ( F 
\  { 0 } )  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B )  =/=  0 ) )
4340, 42sylib 189 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( ( A ^ -u B )  e.  F  /\  ( A ^ -u B
)  =/=  0 ) )
4443simprd 450 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  =/=  0 )
45 neeq1 2558 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x  =/=  0  <->  ( A ^ -u B )  =/=  0 ) )
46 oveq2 6028 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
4746eleq1d 2453 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
4845, 47imbi12d 312 . . . . . . . 8  |-  ( x  =  ( A ^ -u B )  ->  (
( x  =/=  0  ->  ( 1  /  x
)  e.  F )  <-> 
( ( A ^ -u B )  =/=  0  ->  ( 1  /  ( A ^ -u B ) )  e.  F ) ) )
49 expcl2lem.4 . . . . . . . . 9  |-  ( ( x  e.  F  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  F )
5049ex 424 . . . . . . . 8  |-  ( x  e.  F  ->  (
x  =/=  0  -> 
( 1  /  x
)  e.  F ) )
5148, 50vtoclga 2960 . . . . . . 7  |-  ( ( A ^ -u B
)  e.  F  -> 
( ( A ^ -u B )  =/=  0  ->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5241, 44, 51sylc 58 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  F )
5315, 52eqeltrd 2461 . . . . 5  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F )
5453ex 424 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F ) )
557, 54jaod 370 . . 3  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e. 
NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F ) )
561, 55syl5bi 209 . 2  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  F ) )
57563impia 1150 1  |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260    C_ wss 3263   {csn 3757  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928   -ucneg 9224    / cdiv 9609   NNcn 9932   NN0cn0 10153   ZZcz 10214   ^cexp 11309
This theorem is referenced by:  rpexpcl  11327  reexpclz  11328  qexpclz  11329  m1expcl2  11330  expclzlem  11332  1exp  11336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-seq 11251  df-exp 11310
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