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Theorem expcl2lemap 10625
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 9334 . . 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 10624 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 115 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 276 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
8 simpll 527 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sselid 3178 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simplr 528 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A #  0 )
11 simprl 529 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1211recnd 8050 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
13 nnnn0 9250 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1413ad2antll 491 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
15 expineg2 10622 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  -u B  e.  NN0 ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
169, 10, 12, 14, 15syl22anc 1250 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
17 ssrab2 3265 . . . . . . . 8  |-  { z  e.  F  |  z #  0 }  C_  F
18 simpl 109 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A  e.  F  /\  A #  0 ) )
19 breq1 4033 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
z #  0  <->  A #  0
) )
2019elrab 2917 . . . . . . . . . 10  |-  ( A  e.  { z  e.  F  |  z #  0 }  <->  ( A  e.  F  /\  A #  0 ) )
2118, 20sylibr 134 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  { z  e.  F  |  z #  0 }
)
2217, 2sstri 3189 . . . . . . . . . 10  |-  { z  e.  F  |  z #  0 }  C_  CC
2317sseli 3176 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  x  e.  F )
2417sseli 3176 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  y  e.  F )
2523, 24, 3syl2an 289 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y )  e.  F
)
26 breq1 4033 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
z #  0  <->  x #  0
) )
2726elrab 2917 . . . . . . . . . . . . 13  |-  ( x  e.  { z  e.  F  |  z #  0 }  <->  ( x  e.  F  /\  x #  0 ) )
282sseli 3176 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2928anim1i 340 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x #  0 )  ->  (
x  e.  CC  /\  x #  0 ) )
3027, 29sylbi 121 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  ( x  e.  CC  /\  x #  0 ) )
31 breq1 4033 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  (
z #  0  <->  y #  0
) )
3231elrab 2917 . . . . . . . . . . . . 13  |-  ( y  e.  { z  e.  F  |  z #  0 }  <->  ( y  e.  F  /\  y #  0 ) )
332sseli 3176 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3433anim1i 340 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y #  0 )  ->  (
y  e.  CC  /\  y #  0 ) )
3532, 34sylbi 121 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  ( y  e.  CC  /\  y #  0 ) )
36 mulap0 8675 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x #  0 )  /\  ( y  e.  CC  /\  y #  0 ) )  ->  ( x  x.  y ) #  0 )
3730, 35, 36syl2an 289 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y ) #  0 )
38 breq1 4033 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  y )  ->  (
z #  0  <->  ( x  x.  y ) #  0 ) )
3938elrab 2917 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  { z  e.  F  |  z #  0 }  <->  ( ( x  x.  y )  e.  F  /\  ( x  x.  y ) #  0 ) )
4025, 37, 39sylanbrc 417 . . . . . . . . . 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 8611 . . . . . . . . . . 11  |-  1 #  0
42 breq1 4033 . . . . . . . . . . . 12  |-  ( z  =  1  ->  (
z #  0  <->  1 #  0
) )
4342elrab 2917 . . . . . . . . . . 11  |-  ( 1  e.  { z  e.  F  |  z #  0 }  <->  ( 1  e.  F  /\  1 #  0 ) )
444, 41, 43mpbir2an 944 . . . . . . . . . 10  |-  1  e.  { z  e.  F  |  z #  0 }
4522, 40, 44expcllem 10624 . . . . . . . . 9  |-  ( ( A  e.  { z  e.  F  |  z #  0 }  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4621, 14, 45syl2anc 411 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4717, 46sselid 3178 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  F )
48 breq1 4033 . . . . . . . . . 10  |-  ( z  =  ( A ^ -u B )  ->  (
z #  0  <->  ( A ^ -u B ) #  0 ) )
4948elrab 2917 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  { z  e.  F  |  z #  0 }  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B ) #  0 ) )
5046, 49sylib 122 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  (
( A ^ -u B
)  e.  F  /\  ( A ^ -u B
) #  0 ) )
5150simprd 114 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B ) #  0 )
52 breq1 4033 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x #  0  <->  ( A ^ -u B ) #  0 ) )
53 oveq2 5927 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
5453eleq1d 2262 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5552, 54imbi12d 234 . . . . . . . 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 115 . . . . . . . 8  |-  ( x  e.  F  ->  (
x #  0  ->  (
1  /  x )  e.  F ) )
5855, 57vtoclga 2827 . . . . . . 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 2270 . . . . 5  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F )
6160ex 115 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F
) )
627, 61jaod 718 . . 3  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F ) )
631, 62biimtrid 152 . 2  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  ZZ  ->  ( A ^ B )  e.  F ) )
64633impia 1202 1  |-  ( ( A  e.  F  /\  A #  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2164   {crab 2476    C_ wss 3154   class class class wbr 4030  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875    x. cmul 7879   -ucneg 8193   # cap 8602    / cdiv 8693   NNcn 8984   NN0cn0 9243   ZZcz 9320   ^cexp 10612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-exp 10613
This theorem is referenced by:  rpexpcl  10632  reexpclzap  10633  qexpclz  10634  m1expcl2  10635  expclzaplem  10637  1exp  10642
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