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Theorem expcl2lemap 10260
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 9026 . . 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 10259 . . . . . 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 274 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
8 simpll 503 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 3065 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simplr 504 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A #  0 )
11 simprl 505 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1211recnd 7762 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
13 nnnn0 8942 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1413ad2antll 482 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
15 expineg2 10257 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  -u B  e.  NN0 ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
169, 10, 12, 14, 15syl22anc 1202 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
17 ssrab2 3152 . . . . . . . 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 3902 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
z #  0  <->  A #  0
) )
2019elrab 2813 . . . . . . . . . 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 3076 . . . . . . . . . 10  |-  { z  e.  F  |  z #  0 }  C_  CC
2317sseli 3063 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  x  e.  F )
2417sseli 3063 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  y  e.  F )
2523, 24, 3syl2an 287 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y )  e.  F
)
26 breq1 3902 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
z #  0  <->  x #  0
) )
2726elrab 2813 . . . . . . . . . . . . 13  |-  ( x  e.  { z  e.  F  |  z #  0 }  <->  ( x  e.  F  /\  x #  0 ) )
282sseli 3063 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2928anim1i 338 . . . . . . . . . . . . 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 3902 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  (
z #  0  <->  y #  0
) )
3231elrab 2813 . . . . . . . . . . . . 13  |-  ( y  e.  { z  e.  F  |  z #  0 }  <->  ( y  e.  F  /\  y #  0 ) )
332sseli 3063 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3433anim1i 338 . . . . . . . . . . . . 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 8382 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x #  0 )  /\  ( y  e.  CC  /\  y #  0 ) )  ->  ( x  x.  y ) #  0 )
3730, 35, 36syl2an 287 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y ) #  0 )
38 breq1 3902 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  y )  ->  (
z #  0  <->  ( x  x.  y ) #  0 ) )
3938elrab 2813 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  { z  e.  F  |  z #  0 }  <->  ( ( x  x.  y )  e.  F  /\  ( x  x.  y ) #  0 ) )
4025, 37, 39sylanbrc 413 . . . . . . . . . 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 8319 . . . . . . . . . . 11  |-  1 #  0
42 breq1 3902 . . . . . . . . . . . 12  |-  ( z  =  1  ->  (
z #  0  <->  1 #  0
) )
4342elrab 2813 . . . . . . . . . . 11  |-  ( 1  e.  { z  e.  F  |  z #  0 }  <->  ( 1  e.  F  /\  1 #  0 ) )
444, 41, 43mpbir2an 911 . . . . . . . . . 10  |-  1  e.  { z  e.  F  |  z #  0 }
4522, 40, 44expcllem 10259 . . . . . . . . 9  |-  ( ( A  e.  { z  e.  F  |  z #  0 }  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4621, 14, 45syl2anc 408 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4717, 46sseldi 3065 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  F )
48 breq1 3902 . . . . . . . . . 10  |-  ( z  =  ( A ^ -u B )  ->  (
z #  0  <->  ( A ^ -u B ) #  0 ) )
4948elrab 2813 . . . . . . . . 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 3902 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x #  0  <->  ( A ^ -u B ) #  0 ) )
53 oveq2 5750 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
5453eleq1d 2186 . . . . . . . . 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 2726 . . . . . . 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 2194 . . . . 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 691 . . 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 1163 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 682    /\ w3a 947    = wceq 1316    e. wcel 1465   {crab 2397    C_ wss 3041   class class class wbr 3899  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    x. cmul 7593   -ucneg 7902   # cap 8310    / cdiv 8399   NNcn 8684   NN0cn0 8935   ZZcz 9012   ^cexp 10247
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-seqfrec 10174  df-exp 10248
This theorem is referenced by:  rpexpcl  10267  reexpclzap  10268  qexpclz  10269  m1expcl2  10270  expclzaplem  10272  1exp  10277
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