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

Theorem expeq0 12199
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 6304 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21eqeq1d 2459 . . . . 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 6304 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65eqeq1d 2459 . . . . 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 6304 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109eqeq1d 2459 . . . . 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 6304 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413eqeq1d 2459 . . . . 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 12175 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817eqeq1d 2459 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 )  =  0  <->  A  =  0 ) )
19 nnnn0 10823 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 12176 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120eqeq1d 2459 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  x.  A
)  =  0 ) )
22 expcl 12187 . . . . . . . . . 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 10220 . . . . . . . . 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 10574 . 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 1395    e. wcel 1819  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NNcn 10556   NN0cn0 10816   ^cexp 12169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12111  df-exp 12170
This theorem is referenced by:  expne0  12200  0exp  12204  sqeq0  12235  expeq0d  12309  rpexp  14273
  Copyright terms: Public domain W3C validator