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Theorem expap0 10523
Description: Positive integer exponentiation is apart from zero iff its base is apart from zero. That it is easier to prove this first, and then prove expeq0 10524 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." (Remark of [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 5876 . . . . . 6  |-  ( j  =  1  ->  ( A ^ j )  =  ( A ^ 1 ) )
21breq1d 4010 . . . . 5  |-  ( j  =  1  ->  (
( A ^ j
) #  0  <->  ( A ^ 1 ) #  0 ) )
32bibi1d 233 . . . 4  |-  ( j  =  1  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ 1 ) #  0  <-> 
A #  0 ) ) )
43imbi2d 230 . . 3  |-  ( j  =  1  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ 1 ) #  0  <-> 
A #  0 ) ) ) )
5 oveq2 5876 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
65breq1d 4010 . . . . 5  |-  ( j  =  k  ->  (
( A ^ j
) #  0  <->  ( A ^ k ) #  0 ) )
76bibi1d 233 . . . 4  |-  ( j  =  k  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ k ) #  0  <-> 
A #  0 ) ) )
87imbi2d 230 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ k ) #  0  <-> 
A #  0 ) ) ) )
9 oveq2 5876 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
109breq1d 4010 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
) #  0  <->  ( A ^ ( k  +  1 ) ) #  0 ) )
1110bibi1d 233 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) ) )
1211imbi2d 230 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) ) ) )
13 oveq2 5876 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1413breq1d 4010 . . . . 5  |-  ( j  =  N  ->  (
( A ^ j
) #  0  <->  ( A ^ N ) #  0 ) )
1514bibi1d 233 . . . 4  |-  ( j  =  N  ->  (
( ( A ^
j ) #  0  <->  A #  0 )  <->  ( ( A ^ N ) #  0  <-> 
A #  0 ) ) )
1615imbi2d 230 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( ( A ^
j ) #  0  <->  A #  0 ) )  <->  ( A  e.  CC  ->  ( ( A ^ N ) #  0  <-> 
A #  0 ) ) ) )
17 exp1 10499 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1817breq1d 4010 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 1 ) #  0  <->  A #  0
) )
19 nnnn0 9159 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
20 expp1 10500 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2120breq1d 4010 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2221ancoms 268 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  CC )  ->  ( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2319, 22sylan 283 . . . . . . . 8  |-  ( ( k  e.  NN  /\  A  e.  CC )  ->  ( ( A ^
( k  +  1 ) ) #  0  <->  (
( A ^ k
)  x.  A ) #  0 ) )
2423adantr 276 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
( ( A ^
k )  x.  A
) #  0 ) )
25 simplr 528 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  A  e.  CC )
2619ad2antrr 488 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  k  e.  NN0 )
27 expcl 10511 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
2825, 26, 27syl2anc 411 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( A ^
k )  e.  CC )
2928, 25mulap0bd 8590 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( ( A ^ k ) #  0  /\  A #  0 )  <->  ( ( A ^ k )  x.  A ) #  0 ) )
30 anbi1 466 . . . . . . . 8  |-  ( ( ( A ^ k
) #  0  <->  A #  0
)  ->  ( (
( A ^ k
) #  0  /\  A #  0 )  <->  ( A #  0  /\  A #  0 ) ) )
3130adantl 277 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( ( A ^ k ) #  0  /\  A #  0 )  <->  ( A #  0  /\  A #  0 ) ) )
3224, 29, 313bitr2d 216 . . . . . 6  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
( A #  0  /\  A #  0 ) ) )
33 anidm 396 . . . . . 6  |-  ( ( A #  0  /\  A #  0 )  <->  A #  0
)
3432, 33bitrdi 196 . . . . 5  |-  ( ( ( k  e.  NN  /\  A  e.  CC )  /\  ( ( A ^ k ) #  0  <-> 
A #  0 ) )  ->  ( ( A ^ ( k  +  1 ) ) #  0  <-> 
A #  0 ) )
3534exp31 364 . . . 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 8911 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  ( ( A ^ N
) #  0  <->  A #  0
) ) )
3837impcom 125 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N ) #  0  <->  A #  0
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4000  (class class class)co 5868   CCcc 7787   0cc0 7789   1c1 7790    + caddc 7792    x. cmul 7794   # cap 8515   NNcn 8895   NN0cn0 9152   ^cexp 10492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-exp 10493
This theorem is referenced by:  expeq0  10524  abs00ap  11042
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