MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expcl2lem Unicode version

Theorem expcl2lem 11131
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 10053 . . 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 11130 . . . . . 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 3191 . . . . . . 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 8877 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  ->  B  e.  CC )
12 nnnn0 9988 . . . . . . . 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 11128 . . . . . . 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 3316 . . . . . . . 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 3762 . . . . . . . . . 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 3201 . . . . . . . . . 10  |-  ( F 
\  { 0 } )  C_  CC
2116sseli 3189 . . . . . . . . . . . 12  |-  ( x  e.  ( F  \  { 0 } )  ->  x  e.  F
)
2216sseli 3189 . . . . . . . . . . . 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 3762 . . . . . . . . . . . . 13  |-  ( x  e.  ( F  \  { 0 } )  <-> 
( x  e.  F  /\  x  =/=  0
) )
252sseli 3189 . . . . . . . . . . . . . 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 3762 . . . . . . . . . . . . 13  |-  ( y  e.  ( F  \  { 0 } )  <-> 
( y  e.  F  /\  y  =/=  0
) )
292sseli 3189 . . . . . . . . . . . . . 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 9426 . . . . . . . . . . . 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 3762 . . . . . . . . . . 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 8822 . . . . . . . . . . 11  |-  1  =/=  0
37 eldifsn 3762 . . . . . . . . . . 11  |-  ( 1  e.  ( F  \  { 0 } )  <-> 
( 1  e.  F  /\  1  =/=  0
) )
384, 36, 37mpbir2an 886 . . . . . . . . . 10  |-  1  e.  ( F  \  {
0 } )
3920, 35, 38expcllem 11130 . . . . . . . . 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 3191 . . . . . . 7  |-  ( ( ( A  e.  F  /\  A  =/=  0
)  /\  ( B  e.  RR  /\  -u B  e.  NN ) )  -> 
( A ^ -u B
)  e.  F )
42 eldifsn 3762 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( x  =  ( A ^ -u B )  ->  (
x  =/=  0  <->  ( A ^ -u B )  =/=  0 ) )
46 oveq2 5882 . . . . . . . . . 10  |-  ( x  =  ( A ^ -u B )  ->  (
1  /  x )  =  ( 1  / 
( A ^ -u B
) ) )
4746eleq1d 2362 . . . . . . . . 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 2862 . . . . . . 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 2370 . . . . 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120
This theorem is referenced by:  rpexpcl  11138  reexpclz  11139  qexpclz  11140  m1expcl2  11141  expclzlem  11143  1exp  11147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
  Copyright terms: Public domain W3C validator