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Theorem expcl2lem 12281
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 10951 . . 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 12280 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 435 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 466 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  NN0  ->  ( A ^ B
)  e.  F ) )
8 simpll 758 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 3468 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1110recnd 9668 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
12 nnnn0 10876 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1312ad2antll 733 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
14 expneg2 12278 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
159, 11, 13, 14syl3anc 1264 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  =  ( 1  /  ( A ^ -u B ) ) )
16 difss 3598 . . . . . . . 8  |-  ( F 
\  { 0 } )  C_  F
17 simpl 458 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A  e.  F  /\  A  =/=  0
) )
18 eldifsn 4128 . . . . . . . . . 10  |-  ( A  e.  ( F  \  { 0 } )  <-> 
( A  e.  F  /\  A  =/=  0
) )
1917, 18sylibr 215 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  ( F  \  { 0 } ) )
2016, 2sstri 3479 . . . . . . . . . 10  |-  ( F 
\  { 0 } )  C_  CC
2116sseli 3466 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  x  e.  F
)
2216sseli 3466 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  y  e.  F
)
2321, 22, 3syl2an 479 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  F
)
24 eldifsn 4128 . . . . . . . . . . . . 13  |-  ( x  e.  ( F  \  { 0 } )  <-> 
( x  e.  F  /\  x  =/=  0
) )
252sseli 3466 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2625anim1i 570 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x  =/=  0 )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
2724, 26sylbi 198 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
28 eldifsn 4128 . . . . . . . . . . . . 13  |-  ( y  e.  ( F  \  { 0 } )  <-> 
( y  e.  F  /\  y  =/=  0
) )
292sseli 3466 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3029anim1i 570 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y  =/=  0 )  -> 
( y  e.  CC  /\  y  =/=  0 ) )
3128, 30sylbi 198 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  ( y  e.  CC  /\  y  =/=  0 ) )
32 mulne0 10253 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
3327, 31, 32syl2an 479 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  =/=  0
)
34 eldifsn 4128 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  ( F  \  { 0 } )  <-> 
( ( x  x.  y )  e.  F  /\  ( x  x.  y
)  =/=  0 ) )
3523, 33, 34sylanbrc 668 . . . . . . . . . 10  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( F  \  { 0 } ) )
36 ax-1ne0 9607 . . . . . . . . . . 11  |-  1  =/=  0
37 eldifsn 4128 . . . . . . . . . . 11  |-  ( 1  e.  ( F  \  { 0 } )  <-> 
( 1  e.  F  /\  1  =/=  0
) )
384, 36, 37mpbir2an 928 . . . . . . . . . 10  |-  1  e.  ( F  \  {
0 } )
3920, 35, 38expcllem 12280 . . . . . . . . 9  |-  ( ( A  e.  ( F 
\  { 0 } )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  ( F  \  { 0 } ) )
4019, 13, 39syl2anc 665 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  ( F 
\  { 0 } ) )
4116, 40sseldi 3468 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  F )
42 eldifsn 4128 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  ( F 
\  { 0 } )  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B )  =/=  0 ) )
4340, 42sylib 199 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( ( A ^ -u B )  e.  F  /\  ( A ^ -u B
)  =/=  0 ) )
4443simprd 464 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  =/=  0 )
45 neeq1 2712 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x  =/=  0  <->  ( A ^ -u B )  =/=  0 ) )
46 oveq2 6313 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
4746eleq1d 2498 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
4845, 47imbi12d 321 . . . . . . . 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 435 . . . . . . . 8  |-  ( x  e.  F  ->  (
x  =/=  0  -> 
( 1  /  x
)  e.  F ) )
5148, 50vtoclga 3151 . . . . . . 7  |-  ( ( A ^ -u B
)  e.  F  -> 
( ( A ^ -u B )  =/=  0  ->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5241, 44, 51sylc 62 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  F )
5315, 52eqeltrd 2517 . . . . 5  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F )
5453ex 435 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F ) )
557, 54jaod 381 . . 3  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e. 
NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F ) )
561, 55syl5bi 220 . 2  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  F ) )
57563impia 1202 1  |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439    C_ wss 3442   {csn 4002  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   -ucneg 9860    / cdiv 10268   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-seq 12211  df-exp 12270
This theorem is referenced by:  rpexpcl  12288  reexpclz  12289  qexpclz  12290  m1expcl2  12291  expclzlem  12293  1exp  12298
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