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Theorem expcl2lem 11385
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 10287 . . 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 11384 . . . . . 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 3338 . . . . . . 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 9106 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
12 nnnn0 10220 . . . . . . . 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 11382 . . . . . . 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 3466 . . . . . . . 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 3919 . . . . . . . . . 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 3349 . . . . . . . . . 10  |-  ( F 
\  { 0 } )  C_  CC
2116sseli 3336 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  x  e.  F
)
2216sseli 3336 . . . . . . . . . . . 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 3919 . . . . . . . . . . . . 13  |-  ( x  e.  ( F  \  { 0 } )  <-> 
( x  e.  F  /\  x  =/=  0
) )
252sseli 3336 . . . . . . . . . . . . . 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 3919 . . . . . . . . . . . . 13  |-  ( y  e.  ( F  \  { 0 } )  <-> 
( y  e.  F  /\  y  =/=  0
) )
292sseli 3336 . . . . . . . . . . . . . 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 9656 . . . . . . . . . . . 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 3919 . . . . . . . . . . 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 9051 . . . . . . . . . . 11  |-  1  =/=  0
37 eldifsn 3919 . . . . . . . . . . 11  |-  ( 1  e.  ( F  \  { 0 } )  <-> 
( 1  e.  F  /\  1  =/=  0
) )
384, 36, 37mpbir2an 887 . . . . . . . . . 10  |-  1  e.  ( F  \  {
0 } )
3920, 35, 38expcllem 11384 . . . . . . . . 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 3338 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  F )
42 eldifsn 3919 . . . . . . . . 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 2606 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x  =/=  0  <->  ( A ^ -u B )  =/=  0 ) )
46 oveq2 6081 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
4746eleq1d 2501 . . . . . . . . 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 3009 . . . . . . 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 2509 . . . . 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 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   {csn 3806  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987   -ucneg 9284    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ^cexp 11374
This theorem is referenced by:  rpexpcl  11392  reexpclz  11393  qexpclz  11394  m1expcl2  11395  expclzlem  11397  1exp  11401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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