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Theorem expcl2lemap 9432
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 8316 . . 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 9431 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  NN0 )  -> 
( A ^ B
)  e.  F )
65ex 112 . . . . 5  |-  ( A  e.  F  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
76adantr 265 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  NN0  ->  ( A ^ B )  e.  F ) )
8 simpll 489 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  F )
92, 8sseldi 2971 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  CC )
10 simplr 490 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A #  0 )
11 simprl 491 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  RR )
1211recnd 7113 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
13 nnnn0 8246 . . . . . . . 8  |-  ( -u B  e.  NN  ->  -u B  e.  NN0 )
1413ad2antll 468 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  -u B  e.  NN0 )
15 expineg2 9429 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  -u B  e.  NN0 ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
169, 10, 12, 14, 15syl22anc 1147 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  =  ( 1  /  ( A ^ -u B ) ) )
17 ssrab2 3053 . . . . . . . 8  |-  { z  e.  F  |  z #  0 }  C_  F
18 simpl 106 . . . . . . . . . 10  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A  e.  F  /\  A #  0 ) )
19 breq1 3795 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
z #  0  <->  A #  0
) )
2019elrab 2721 . . . . . . . . . 10  |-  ( A  e.  { z  e.  F  |  z #  0 }  <->  ( A  e.  F  /\  A #  0 ) )
2118, 20sylibr 141 . . . . . . . . 9  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  A  e.  { z  e.  F  |  z #  0 }
)
2217, 2sstri 2982 . . . . . . . . . 10  |-  { z  e.  F  |  z #  0 }  C_  CC
2317sseli 2969 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  x  e.  F )
2417sseli 2969 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  y  e.  F )
2523, 24, 3syl2an 277 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y )  e.  F
)
26 breq1 3795 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
z #  0  <->  x #  0
) )
2726elrab 2721 . . . . . . . . . . . . 13  |-  ( x  e.  { z  e.  F  |  z #  0 }  <->  ( x  e.  F  /\  x #  0 ) )
282sseli 2969 . . . . . . . . . . . . . 14  |-  ( x  e.  F  ->  x  e.  CC )
2928anim1i 327 . . . . . . . . . . . . 13  |-  ( ( x  e.  F  /\  x #  0 )  ->  (
x  e.  CC  /\  x #  0 ) )
3027, 29sylbi 118 . . . . . . . . . . . 12  |-  ( x  e.  { z  e.  F  |  z #  0 }  ->  ( x  e.  CC  /\  x #  0 ) )
31 breq1 3795 . . . . . . . . . . . . . 14  |-  ( z  =  y  ->  (
z #  0  <->  y #  0
) )
3231elrab 2721 . . . . . . . . . . . . 13  |-  ( y  e.  { z  e.  F  |  z #  0 }  <->  ( y  e.  F  /\  y #  0 ) )
332sseli 2969 . . . . . . . . . . . . . 14  |-  ( y  e.  F  ->  y  e.  CC )
3433anim1i 327 . . . . . . . . . . . . 13  |-  ( ( y  e.  F  /\  y #  0 )  ->  (
y  e.  CC  /\  y #  0 ) )
3532, 34sylbi 118 . . . . . . . . . . . 12  |-  ( y  e.  { z  e.  F  |  z #  0 }  ->  ( y  e.  CC  /\  y #  0 ) )
36 mulap0 7709 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x #  0 )  /\  ( y  e.  CC  /\  y #  0 ) )  ->  ( x  x.  y ) #  0 )
3730, 35, 36syl2an 277 . . . . . . . . . . 11  |-  ( ( x  e.  { z  e.  F  |  z #  0 }  /\  y  e.  { z  e.  F  |  z #  0 }
)  ->  ( x  x.  y ) #  0 )
38 breq1 3795 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  y )  ->  (
z #  0  <->  ( x  x.  y ) #  0 ) )
3938elrab 2721 . . . . . . . . . . 11  |-  ( ( x  x.  y )  e.  { z  e.  F  |  z #  0 }  <->  ( ( x  x.  y )  e.  F  /\  ( x  x.  y ) #  0 ) )
4025, 37, 39sylanbrc 402 . . . . . . . . . 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 7655 . . . . . . . . . . 11  |-  1 #  0
42 breq1 3795 . . . . . . . . . . . 12  |-  ( z  =  1  ->  (
z #  0  <->  1 #  0
) )
4342elrab 2721 . . . . . . . . . . 11  |-  ( 1  e.  { z  e.  F  |  z #  0 }  <->  ( 1  e.  F  /\  1 #  0 ) )
444, 41, 43mpbir2an 860 . . . . . . . . . 10  |-  1  e.  { z  e.  F  |  z #  0 }
4522, 40, 44expcllem 9431 . . . . . . . . 9  |-  ( ( A  e.  { z  e.  F  |  z #  0 }  /\  -u B  e.  NN0 )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4621, 14, 45syl2anc 397 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  { z  e.  F  |  z #  0 } )
4717, 46sseldi 2971 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B )  e.  F )
48 breq1 3795 . . . . . . . . . 10  |-  ( z  =  ( A ^ -u B )  ->  (
z #  0  <->  ( A ^ -u B ) #  0 ) )
4948elrab 2721 . . . . . . . . 9  |-  ( ( A ^ -u B
)  e.  { z  e.  F  |  z #  0 }  <->  ( ( A ^ -u B )  e.  F  /\  ( A ^ -u B ) #  0 ) )
5046, 49sylib 131 . . . . . . . 8  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  (
( A ^ -u B
)  e.  F  /\  ( A ^ -u B
) #  0 ) )
5150simprd 111 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ -u B ) #  0 )
52 breq1 3795 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x #  0  <->  ( A ^ -u B ) #  0 ) )
53 oveq2 5548 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
5453eleq1d 2122 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
( 1  /  x
)  e.  F  <->  ( 1  /  ( A ^ -u B ) )  e.  F ) )
5552, 54imbi12d 227 . . . . . . . 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 112 . . . . . . . 8  |-  ( x  e.  F  ->  (
x #  0  ->  (
1  /  x )  e.  F ) )
5855, 57vtoclga 2636 . . . . . . 7  |-  ( ( A ^ -u B
)  e.  F  -> 
( ( A ^ -u B ) #  0  -> 
( 1  /  ( A ^ -u B ) )  e.  F ) )
5947, 51, 58sylc 60 . . . . . 6  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  (
1  /  ( A ^ -u B ) )  e.  F )
6016, 59eqeltrd 2130 . . . . 5  |-  ( ( ( A  e.  F  /\  A #  0 )  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F )
6160ex 112 . . . 4  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  RR  /\  -u B  e.  NN )  ->  ( A ^ B )  e.  F
) )
627, 61jaod 647 . . 3  |-  ( ( A  e.  F  /\  A #  0 )  ->  (
( B  e.  NN0  \/  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  ( A ^ B )  e.  F ) )
631, 62syl5bi 145 . 2  |-  ( ( A  e.  F  /\  A #  0 )  ->  ( B  e.  ZZ  ->  ( A ^ B )  e.  F ) )
64633impia 1112 1  |-  ( ( A  e.  F  /\  A #  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   {crab 2327    C_ wss 2945   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    x. cmul 6952   -ucneg 7246   # cap 7646    / cdiv 7725   NNcn 7990   NN0cn0 8239   ZZcz 8302   ^cexp 9419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376  df-iexp 9420
This theorem is referenced by:  rpexpcl  9439  reexpclzap  9440  qexpclz  9441  m1expcl2  9442  expclzaplem  9444  1exp  9449
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