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Theorem expeq0 11132
Description: Natural number 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 5866 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21eqeq1d 2291 . . . . 5  |-  ( j  =  1  ->  (
( A ^ j
)  =  0  <->  ( A ^ 1 )  =  0 ) )
32bibi1d 310 . . . 4  |-  ( j  =  1  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
1 )  =  0  <-> 
A  =  0 ) ) )
43imbi2d 307 . . 3  |-  ( j  =  1  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ 1 )  =  0  <->  A  =  0
) ) ) )
5 oveq2 5866 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65eqeq1d 2291 . . . . 5  |-  ( j  =  k  ->  (
( A ^ j
)  =  0  <->  ( A ^ k )  =  0 ) )
76bibi1d 310 . . . 4  |-  ( j  =  k  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
k )  =  0  <-> 
A  =  0 ) ) )
87imbi2d 307 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ k )  =  0  <->  A  =  0
) ) ) )
9 oveq2 5866 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109eqeq1d 2291 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  =  0  <->  ( A ^ ( k  +  1 ) )  =  0 ) )
1110bibi1d 310 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
A  =  0 ) ) )
1211imbi2d 307 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
13 oveq2 5866 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413eqeq1d 2291 . . . . 5  |-  ( j  =  N  ->  (
( A ^ j
)  =  0  <->  ( A ^ N )  =  0 ) )
1514bibi1d 310 . . . 4  |-  ( j  =  N  ->  (
( ( A ^
j )  =  0  <-> 
A  =  0 )  <-> 
( ( A ^ N )  =  0  <-> 
A  =  0 ) ) )
1615imbi2d 307 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( ( A ^
j )  =  0  <-> 
A  =  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ N )  =  0  <->  A  =  0
) ) ) )
17 exp1 11109 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817eqeq1d 2291 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 )  =  0  <->  A  =  0 ) )
19 nnnn0 9972 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 11110 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120eqeq1d 2291 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  x.  A
)  =  0 ) )
22 expcl 11121 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
23 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
2422, 23mul0ord 9418 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( A ^ k )  x.  A )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
2521, 24bitrd 244 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
2619, 25sylan2 460 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  =  0  \/  A  =  0 ) ) )
27 bi1 178 . . . . . . . . 9  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( A ^
k )  =  0  ->  A  =  0 ) )
28 idd 21 . . . . . . . . 9  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( A  =  0  ->  A  =  0 ) )
2927, 28jaod 369 . . . . . . . 8  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( ( A ^ k )  =  0  \/  A  =  0 )  ->  A  =  0 ) )
30 olc 373 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ k
)  =  0  \/  A  =  0 ) )
3129, 30impbid1 194 . . . . . . 7  |-  ( ( ( A ^ k
)  =  0  <->  A  =  0 )  -> 
( ( ( A ^ k )  =  0  \/  A  =  0 )  <->  A  = 
0 ) )
3226, 31sylan9bb 680 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN )  /\  ( ( A ^ k )  =  0  <->  A  =  0
) )  ->  (
( A ^ (
k  +  1 ) )  =  0  <->  A  =  0 ) )
3332exp31 587 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN  ->  ( ( ( A ^
k )  =  0  <-> 
A  =  0 )  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
3433com12 27 . . . 4  |-  ( k  e.  NN  ->  ( A  e.  CC  ->  ( ( ( A ^
k )  =  0  <-> 
A  =  0 )  ->  ( ( A ^ ( k  +  1 ) )  =  0  <->  A  =  0
) ) ) )
3534a2d 23 . . 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 9764 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  ( ( A ^ N
)  =  0  <->  A  =  0 ) ) )
3736impcom 419 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  expne0  11133  0exp  11137  sqeq0  11168  expeq0d  11241  rpexp  12799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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