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Theorem expeq0 12151
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 6283 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21eqeq1d 2462 . . . . 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 6283 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65eqeq1d 2462 . . . . 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 6283 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109eqeq1d 2462 . . . . 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 6283 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413eqeq1d 2462 . . . . 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 12128 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817eqeq1d 2462 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 )  =  0  <->  A  =  0 ) )
19 nnnn0 10791 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 12129 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120eqeq1d 2462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) )  =  0  <-> 
( ( A ^
k )  x.  A
)  =  0 ) )
22 expcl 12140 . . . . . . . . . 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 10188 . . . . . . . . 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 10543 . 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 1374    e. wcel 1762  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NNcn 10525   NN0cn0 10784   ^cexp 12122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-seq 12064  df-exp 12123
This theorem is referenced by:  expne0  12152  0exp  12156  sqeq0  12187  expeq0d  12261  rpexp  14109
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