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Theorem expcl2lem 11115
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 10037 . . 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 11114 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 423 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 451 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  NN0  ->  ( A ^ B
)  e.  F ) )
8 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 3178 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simprl 732 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1110recnd 8861 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
12 nnnn0 9972 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1312ad2antll 709 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
14 expneg2 11112 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u B  e.  NN0 )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
159, 11, 13, 14syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  =  ( 1  /  ( A ^ -u B ) ) )
16 difss 3303 . . . . . . . 8  |-  ( F 
\  { 0 } )  C_  F
17 simpl 443 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A  e.  F  /\  A  =/=  0
) )
18 eldifsn 3749 . . . . . . . . . 10  |-  ( A  e.  ( F  \  { 0 } )  <-> 
( A  e.  F  /\  A  =/=  0
) )
1917, 18sylibr 203 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  ( F  \  { 0 } ) )
2016, 2sstri 3188 . . . . . . . . . 10  |-  ( F 
\  { 0 } )  C_  CC
2116sseli 3176 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  x  e.  F
)
2216sseli 3176 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  y  e.  F
)
2321, 22, 3syl2an 463 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  F
)
24 eldifsn 3749 . . . . . . . . . . . . 13  |-  ( x  e.  ( F  \  { 0 } )  <-> 
( x  e.  F  /\  x  =/=  0
) )
252sseli 3176 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2625anim1i 551 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x  =/=  0 )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
2724, 26sylbi 187 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
28 eldifsn 3749 . . . . . . . . . . . . 13  |-  ( y  e.  ( F  \  { 0 } )  <-> 
( y  e.  F  /\  y  =/=  0
) )
292sseli 3176 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3029anim1i 551 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y  =/=  0 )  -> 
( y  e.  CC  /\  y  =/=  0 ) )
3128, 30sylbi 187 . . . . . . . . . . . 12  |-  ( y  e.  ( F  \  { 0 } )  ->  ( y  e.  CC  /\  y  =/=  0 ) )
32 mulne0 9410 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
3327, 31, 32syl2an 463 . . . . . . . . . . 11  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  =/=  0
)
34 eldifsn 3749 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  ( F  \  { 0 } )  <-> 
( ( x  x.  y )  e.  F  /\  ( x  x.  y
)  =/=  0 ) )
3523, 33, 34sylanbrc 645 . . . . . . . . . 10  |-  ( ( x  e.  ( F 
\  { 0 } )  /\  y  e.  ( F  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( F  \  { 0 } ) )
36 ax-1ne0 8806 . . . . . . . . . . 11  |-  1  =/=  0
37 eldifsn 3749 . . . . . . . . . . 11  |-  ( 1  e.  ( F  \  { 0 } )  <-> 
( 1  e.  F  /\  1  =/=  0
) )
384, 36, 37mpbir2an 886 . . . . . . . . . 10  |-  1  e.  ( F  \  {
0 } )
3920, 35, 38expcllem 11114 . . . . . . . . 9  |-  ( ( A  e.  ( F 
\  { 0 } )  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  ( F  \  { 0 } ) )
4019, 13, 39syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  ( F 
\  { 0 } ) )
4116, 40sseldi 3178 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  F )
42 eldifsn 3749 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  ( F 
\  { 0 } )  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B )  =/=  0 ) )
4340, 42sylib 188 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( ( A ^ -u B )  e.  F  /\  ( A ^ -u B
)  =/=  0 ) )
4443simprd 449 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  =/=  0 )
45 neeq1 2454 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x  =/=  0  <->  ( A ^ -u B )  =/=  0 ) )
46 oveq2 5866 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
4746eleq1d 2349 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
4845, 47imbi12d 311 . . . . . . . 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 423 . . . . . . . 8  |-  ( x  e.  F  ->  (
x  =/=  0  -> 
( 1  /  x
)  e.  F ) )
5148, 50vtoclga 2849 . . . . . . 7  |-  ( ( A ^ -u B
)  e.  F  -> 
( ( A ^ -u B )  =/=  0  ->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5241, 44, 51sylc 56 . . . . . 6  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( 1  /  ( A ^ -u B ) )  e.  F )
5315, 52eqeltrd 2357 . . . . 5  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F )
5453ex 423 . . . 4  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F ) )
557, 54jaod 369 . . 3  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( ( B  e. 
NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ B
)  e.  F ) )
561, 55syl5bi 208 . 2  |-  ( ( A  e.  F  /\  A  =/=  0 )  -> 
( B  e.  ZZ  ->  ( A ^ B
)  e.  F ) )
57563impia 1148 1  |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   -ucneg 9038    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104
This theorem is referenced by:  rpexpcl  11122  reexpclz  11123  qexpclz  11124  m1expcl2  11125  expclzlem  11127  1exp  11131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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