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Theorem expcl2lemap 10245
Description: Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
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
expcl2lemap.4  |-  ( ( x  e.  F  /\  x #  0 )  ->  (
1  /  x )  e.  F )
Assertion
Ref Expression
expcl2lemap  |-  ( ( 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 expcl2lemap
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elznn0nn 9019 . . 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 10244 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 114 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 272 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
8 simpll 501 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 3063 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simplr 502 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A #  0 )
11 simprl 503 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1211recnd 7758 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
13 nnnn0 8935 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1413ad2antll 480 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
15 expineg2 10242 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  -u B  e.  NN0 ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
169, 10, 12, 14, 15syl22anc 1200 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
17 ssrab2 3150 . . . . . . . 8  |-  { z  e.  F  |  z #  0 }  C_  F
18 simpl 108 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A  e.  F  /\  A #  0 ) )
19 breq1 3900 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
z #  0  <->  A #  0
) )
2019elrab 2811 . . . . . . . . . 10  |-  ( A  e.  { z  e.  F  |  z #  0 }  <->  ( A  e.  F  /\  A #  0 ) )
2118, 20sylibr 133 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  { z  e.  F  |  z #  0 }
)
2217, 2sstri 3074 . . . . . . . . . 10  |-  { z  e.  F  |  z #  0 }  C_  CC
2317sseli 3061 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  x  e.  F )
2417sseli 3061 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  y  e.  F )
2523, 24, 3syl2an 285 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y )  e.  F
)
26 breq1 3900 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
z #  0  <->  x #  0
) )
2726elrab 2811 . . . . . . . . . . . . 13  |-  ( x  e.  { z  e.  F  |  z #  0 }  <->  ( x  e.  F  /\  x #  0 ) )
282sseli 3061 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2928anim1i 336 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x #  0 )  ->  (
x  e.  CC  /\  x #  0 ) )
3027, 29sylbi 120 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  ( x  e.  CC  /\  x #  0 ) )
31 breq1 3900 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  (
z #  0  <->  y #  0
) )
3231elrab 2811 . . . . . . . . . . . . 13  |-  ( y  e.  { z  e.  F  |  z #  0 }  <->  ( y  e.  F  /\  y #  0 ) )
332sseli 3061 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3433anim1i 336 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y #  0 )  ->  (
y  e.  CC  /\  y #  0 ) )
3532, 34sylbi 120 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  ( y  e.  CC  /\  y #  0 ) )
36 mulap0 8375 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x #  0 )  /\  ( y  e.  CC  /\  y #  0 ) )  ->  ( x  x.  y ) #  0 )
3730, 35, 36syl2an 285 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y ) #  0 )
38 breq1 3900 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  y )  ->  (
z #  0  <->  ( x  x.  y ) #  0 ) )
3938elrab 2811 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  { z  e.  F  |  z #  0 }  <->  ( ( x  x.  y )  e.  F  /\  ( x  x.  y ) #  0 ) )
4025, 37, 39sylanbrc 411 . . . . . . . . . 10  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y )  e.  {
z  e.  F  | 
z #  0 } )
41 1ap0 8315 . . . . . . . . . . 11  |-  1 #  0
42 breq1 3900 . . . . . . . . . . . 12  |-  ( z  =  1  ->  (
z #  0  <->  1 #  0
) )
4342elrab 2811 . . . . . . . . . . 11  |-  ( 1  e.  { z  e.  F  |  z #  0 }  <->  ( 1  e.  F  /\  1 #  0 ) )
444, 41, 43mpbir2an 909 . . . . . . . . . 10  |-  1  e.  { z  e.  F  |  z #  0 }
4522, 40, 44expcllem 10244 . . . . . . . . 9  |-  ( ( A  e.  { z  e.  F  |  z #  0 }  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4621, 14, 45syl2anc 406 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4717, 46sseldi 3063 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  F )
48 breq1 3900 . . . . . . . . . 10  |-  ( z  =  ( A ^ -u B )  ->  (
z #  0  <->  ( A ^ -u B ) #  0 ) )
4948elrab 2811 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  { z  e.  F  |  z #  0 }  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B ) #  0 ) )
5046, 49sylib 121 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  (
( A ^ -u B
)  e.  F  /\  ( A ^ -u B
) #  0 ) )
5150simprd 113 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B ) #  0 )
52 breq1 3900 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x #  0  <->  ( A ^ -u B ) #  0 ) )
53 oveq2 5748 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
5453eleq1d 2184 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5552, 54imbi12d 233 . . . . . . . 8  |-  ( x  =  ( A ^ -u B )  ->  (
( x #  0  -> 
( 1  /  x
)  e.  F )  <-> 
( ( A ^ -u B ) #  0  -> 
( 1  /  ( A ^ -u B ) )  e.  F ) ) )
56 expcl2lemap.4 . . . . . . . . 9  |-  ( ( x  e.  F  /\  x #  0 )  ->  (
1  /  x )  e.  F )
5756ex 114 . . . . . . . 8  |-  ( x  e.  F  ->  (
x #  0  ->  (
1  /  x )  e.  F ) )
5855, 57vtoclga 2724 . . . . . . 7  |-  ( ( A ^ -u B
)  e.  F  -> 
( ( A ^ -u B ) #  0  -> 
( 1  /  ( A ^ -u B ) )  e.  F ) )
5947, 51, 58sylc 62 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  (
1  /  ( A ^ -u B ) )  e.  F )
6016, 59eqeltrd 2192 . . . . 5  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F )
6160ex 114 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F
) )
627, 61jaod 689 . . 3  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F ) )
631, 62syl5bi 151 . 2  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  ZZ  ->  ( A ^ B )  e.  F ) )
64633impia 1161 1  |-  ( ( A  e.  F  /\  A #  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   {crab 2395    C_ wss 3039   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    x. cmul 7589   -ucneg 7898   # cap 8306    / cdiv 8392   NNcn 8677   NN0cn0 8928   ZZcz 9005   ^cexp 10232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-n0 8929  df-z 9006  df-uz 9276  df-seqfrec 10159  df-exp 10233
This theorem is referenced by:  rpexpcl  10252  reexpclzap  10253  qexpclz  10254  m1expcl2  10255  expclzaplem  10257  1exp  10262
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