MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expeq0 Structured version   Unicode version

Theorem expeq0 11997
Description: Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
Assertion
Ref Expression
expeq0  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )

Proof of Theorem expeq0
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6200 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21eqeq1d 2453 . . . . 5  |-  ( j  =  1  ->  (
( A ^ j
)  =  0  <->  ( A ^ 1 )  =  0 ) )
32bibi1d 319 . . . 4  |-  ( j  =  1  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
1 )  =  0  <-> 
A  =  0 ) ) )
43imbi2d 316 . . 3  |-  ( j  =  1  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ 1 )  =  0  <->  A  =  0
) ) ) )
5 oveq2 6200 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65eqeq1d 2453 . . . . 5  |-  ( j  =  k  ->  (
( A ^ j
)  =  0  <->  ( A ^ k )  =  0 ) )
76bibi1d 319 . . . 4  |-  ( j  =  k  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
k )  =  0  <-> 
A  =  0 ) ) )
87imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ k )  =  0  <->  A  =  0
) ) ) )
9 oveq2 6200 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109eqeq1d 2453 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  =  0  <->  ( A ^ ( k  +  1 ) )  =  0 ) )
1110bibi1d 319 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
A  =  0 ) ) )
1211imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
13 oveq2 6200 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413eqeq1d 2453 . . . . 5  |-  ( j  =  N  ->  (
( A ^ j
)  =  0  <->  ( A ^ N )  =  0 ) )
1514bibi1d 319 . . . 4  |-  ( j  =  N  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^ N )  =  0  <-> 
A  =  0 ) ) )
1615imbi2d 316 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ N )  =  0  <->  A  =  0
) ) ) )
17 exp1 11974 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817eqeq1d 2453 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 )  =  0  <->  A  =  0 ) )
19 nnnn0 10689 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 11975 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120eqeq1d 2453 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  x.  A
)  =  0 ) )
22 expcl 11986 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
23 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
2422, 23mul0ord 10089 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( A ^ k )  x.  A )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
2521, 24bitrd 253 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
2619, 25sylan2 474 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
27 bi1 186 . . . . . . . . 9  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( A ^
k )  =  0  ->  A  =  0 ) )
28 idd 24 . . . . . . . . 9  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( A  =  0  ->  A  =  0 ) )
2927, 28jaod 380 . . . . . . . 8  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( ( A ^ k )  =  0  \/  A  =  0 )  ->  A  =  0 ) )
30 olc 384 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ k
)  =  0  \/  A  =  0 ) )
3129, 30impbid1 203 . . . . . . 7  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( ( A ^ k )  =  0  \/  A  =  0 )  <->  A  = 
0 ) )
3226, 31sylan9bb 699 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN )  /\  ( ( A ^ k )  =  0  <->  A  =  0
) )  ->  (
( A ^ (
k  +  1 ) )  =  0  <->  A  =  0 ) )
3332exp31 604 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN  ->  ( ( ( A ^
k )  =  0  <-> 
A  =  0 )  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
3433com12 31 . . . 4  |-  ( k  e.  NN  ->  ( A  e.  CC  ->  ( ( ( A ^
k )  =  0  <-> 
A  =  0 )  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
3534a2d 26 . . 3  |-  ( k  e.  NN  ->  (
( A  e.  CC  ->  ( ( A ^
k )  =  0  <-> 
A  =  0 ) )  ->  ( A  e.  CC  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
364, 8, 12, 16, 18, 35nnind 10443 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  ( ( A ^ N
)  =  0  <->  A  =  0 ) ) )
3736impcom 430 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6192   CCcc 9383   0cc0 9385   1c1 9386    + caddc 9388    x. cmul 9390   NNcn 10425   NN0cn0 10682   ^cexp 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-seq 11910  df-exp 11969
This theorem is referenced by:  expne0  11998  0exp  12002  sqeq0  12033  expeq0d  12107  rpexp  13910
  Copyright terms: Public domain W3C validator