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Theorem expeq0 11877
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 6088 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21eqeq1d 2441 . . . . 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 6088 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65eqeq1d 2441 . . . . 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 6088 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109eqeq1d 2441 . . . . 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 6088 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413eqeq1d 2441 . . . . 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 11854 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817eqeq1d 2441 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 )  =  0  <->  A  =  0 ) )
19 nnnn0 10573 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 11855 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120eqeq1d 2441 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  x.  A
)  =  0 ) )
22 expcl 11866 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
23 simpl 454 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
2422, 23mul0ord 9973 . . . . . . . . 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 471 . . . . . . 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 692 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN )  /\  ( ( A ^ k )  =  0  <->  A  =  0
) )  ->  (
( A ^ (
k  +  1 ) )  =  0  <->  A  =  0 ) )
3332exp31 599 . . . . 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 10327 . 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 1362    e. wcel 1755  (class class class)co 6080   CCcc 9267   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   NNcn 10309   NN0cn0 10566   ^cexp 11848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-n0 10567  df-z 10634  df-uz 10849  df-seq 11790  df-exp 11849
This theorem is referenced by:  expne0  11878  0exp  11882  sqeq0  11913  expeq0d  11987  rpexp  13788
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