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Theorem expap0 9603
Description: Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9604 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
expap0  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N ) #  0  <->  A #  0
) )

Proof of Theorem expap0
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5551 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21breq1d 3803 . . . . 5  |-  ( j  =  1  ->  (
( A ^ j
) #  0  <->  ( A ^ 1 ) #  0 ) )
32bibi1d 231 . . . 4  |-  ( j  =  1  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ 1 ) #  0  <-> 
A #  0 ) ) )
43imbi2d 228 . . 3  |-  ( j  =  1  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ 1 ) #  0  <-> 
A #  0 ) ) ) )
5 oveq2 5551 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65breq1d 3803 . . . . 5  |-  ( j  =  k  ->  (
( A ^ j
) #  0  <->  ( A ^ k ) #  0 ) )
76bibi1d 231 . . . 4  |-  ( j  =  k  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ k ) #  0  <-> 
A #  0 ) ) )
87imbi2d 228 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ k ) #  0  <-> 
A #  0 ) ) ) )
9 oveq2 5551 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109breq1d 3803 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
) #  0  <->  ( A ^ ( k  +  1 ) ) #  0 ) )
1110bibi1d 231 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) ) )
1211imbi2d 228 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) ) ) )
13 oveq2 5551 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413breq1d 3803 . . . . 5  |-  ( j  =  N  ->  (
( A ^ j
) #  0  <->  ( A ^ N ) #  0 ) )
1514bibi1d 231 . . . 4  |-  ( j  =  N  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ N ) #  0  <-> 
A #  0 ) ) )
1615imbi2d 228 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ N ) #  0  <-> 
A #  0 ) ) ) )
17 exp1 9579 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817breq1d 3803 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 ) #  0  <->  A #  0
) )
19 nnnn0 8362 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 9580 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120breq1d 3803 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2221ancoms 264 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  CC )  ->  ( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2319, 22sylan 277 . . . . . . . 8  |-  ( ( k  e.  NN  /\  A  e.  CC )  ->  ( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2423adantr 270 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
( ( A ^
k )  x.  A
) #  0 ) )
25 simplr 497 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  A  e.  CC )
2619ad2antrr 472 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  k  e.  NN0 )
27 expcl 9591 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
2825, 26, 27syl2anc 403 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( A ^
k )  e.  CC )
2928, 25mulap0bd 7814 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( ( A ^ k ) #  0  /\  A #  0 )  <->  ( ( A ^ k )  x.  A ) #  0 ) )
30 anbi1 454 . . . . . . . 8  |-  ( ( ( A ^ k
) #  0  <->  A #  0
)  ->  ( (
( A ^ k
) #  0  /\  A #  0 )  <->  ( A #  0  /\  A #  0 ) ) )
3130adantl 271 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( ( A ^ k ) #  0  /\  A #  0 )  <->  ( A #  0  /\  A #  0 ) ) )
3224, 29, 313bitr2d 214 . . . . . 6  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
( A #  0  /\  A #  0 ) ) )
33 anidm 388 . . . . . 6  |-  ( ( A #  0  /\  A #  0 )  <->  A #  0
)
3432, 33syl6bb 194 . . . . 5  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) )
3534exp31 356 . . . 4  |-  ( k  e.  NN  ->  ( A  e.  CC  ->  ( ( ( A ^
k ) #  0  <->  A #  0 )  ->  (
( A ^ (
k  +  1 ) ) #  0  <->  A #  0
) ) ) )
3635a2d 26 . . 3  |-  ( k  e.  NN  ->  (
( A  e.  CC  ->  ( ( A ^
k ) #  0  <->  A #  0 ) )  -> 
( A  e.  CC  ->  ( ( A ^
( k  +  1 ) ) #  0  <->  A #  0 ) ) ) )
374, 8, 12, 16, 18, 36nnind 8122 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  ( ( A ^ N
) #  0  <->  A #  0
) ) )
3837impcom 123 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N ) #  0  <->  A #  0
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   # cap 7748   NNcn 8106   NN0cn0 8355   ^cexp 9572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573
This theorem is referenced by:  expeq0  9604  abs00ap  10086
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