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Theorem expivallem 9574
Description: Lemma for expival 9575. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
expivallem  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) #  0 )

Proof of Theorem expivallem
Dummy variables  k  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5209 . . . . . 6  |-  ( n  =  1  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  1 )
)
21breq1d 3803 . . . . 5  |-  ( n  =  1  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  n
) #  0  <->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  1 ) #  0 ) )
32imbi2d 228 . . . 4  |-  ( n  =  1  ->  (
( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n ) #  0 )  <->  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  1 ) #  0 ) ) )
4 fveq2 5209 . . . . . 6  |-  ( n  =  k  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k )
)
54breq1d 3803 . . . . 5  |-  ( n  =  k  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  n
) #  0  <->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k ) #  0 ) )
65imbi2d 228 . . . 4  |-  ( n  =  k  ->  (
( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n ) #  0 )  <->  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k ) #  0 ) ) )
7 fveq2 5209 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) )
87breq1d 3803 . . . . 5  |-  ( n  =  ( k  +  1 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  n
) #  0  <->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) #  0 ) )
98imbi2d 228 . . . 4  |-  ( n  =  ( k  +  1 )  ->  (
( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n ) #  0 )  <->  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) #  0 ) ) )
10 fveq2 5209 . . . . . 6  |-  ( n  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )
)
1110breq1d 3803 . . . . 5  |-  ( n  =  N  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  n
) #  0  <->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) #  0 ) )
1211imbi2d 228 . . . 4  |-  ( n  =  N  ->  (
( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  n ) #  0 )  <->  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) #  0 ) ) )
13 simpr 108 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0 )  ->  A #  0 )
14 1zzd 8459 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0 )  ->  1  e.  ZZ )
15 elnnuz 8736 . . . . . . . . . . 11  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
16 fvconst2g 5407 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
1715, 16sylan2br 282 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  x )  =  A )
1817adantlr 461 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  x )  =  A )
19 simpll 496 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  x  e.  ( ZZ>= ` 
1 ) )  ->  A  e.  CC )
2018, 19eqeltrd 2156 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  x )  e.  CC )
21 mulcl 7162 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2221adantl 271 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
2314, 20, 22iseq1 9533 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  1 )  =  ( ( NN 
X.  { A }
) `  1 )
)
24 1nn 8117 . . . . . . . . 9  |-  1  e.  NN
25 fvconst2g 5407 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2624, 25mpan2 416 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  A )
2726adantr 270 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( NN  X.  { A } ) `  1
)  =  A )
2823, 27eqtrd 2114 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  1 )  =  A )
2928breq1d 3803 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  1
) #  0  <->  A #  0
) )
3013, 29mpbird 165 . . . 4  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  1 ) #  0 )
31 simpl 107 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  k  e.  NN )
32 elnnuz 8736 . . . . . . . . . . 11  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
3331, 32sylib 120 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  k  e.  (
ZZ>= `  1 ) )
3433adantr 270 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  k  e.  ( ZZ>= `  1 )
)
3520adantll 460 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  x  e.  (
ZZ>= `  1 ) )  ->  ( ( NN 
X.  { A }
) `  x )  e.  CC )
3635adantlr 461 . . . . . . . . 9  |-  ( ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k
) #  0 )  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  x )  e.  CC )
3721adantl 271 . . . . . . . . 9  |-  ( ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k
) #  0 )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
3834, 36, 37iseqcl 9537 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  e.  CC )
39 simplrl 502 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  A  e.  CC )
40 simpr 108 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k ) #  0 )
41 simplrr 503 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  A #  0 )
4238, 39, 40, 41mulap0d 7815 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k
)  x.  A ) #  0 )
4321adantl 271 . . . . . . . . . . 11  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
4433, 35, 43iseqp1 9538 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  x.  ( ( NN  X.  { A } ) `  ( k  +  1 ) ) ) )
45 simprl 498 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  A  e.  CC )
4631peano2nnd 8121 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  ( k  +  1 )  e.  NN )
47 fvconst2g 5407 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
4845, 46, 47syl2anc 403 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
4948oveq2d 5559 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  x.  ( ( NN  X.  { A } ) `  ( k  +  1 ) ) )  =  ( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k )  x.  A ) )
5044, 49eqtrd 2114 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  x.  A ) )
5150breq1d 3803 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) #  0  <->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  x.  A ) #  0 ) )
5251adantr 270 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  (
k  +  1 ) ) #  0  <->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k )  x.  A ) #  0 ) )
5342, 52mpbird 165 . . . . . 6  |-  ( ( ( k  e.  NN  /\  ( A  e.  CC  /\  A #  0 ) )  /\  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) #  0 )
5453exp31 356 . . . . 5  |-  ( k  e.  NN  ->  (
( A  e.  CC  /\  A #  0 )  -> 
( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  k ) #  0  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  ( k  +  1 ) ) #  0 ) ) )
5554a2d 26 . . . 4  |-  ( k  e.  NN  ->  (
( ( A  e.  CC  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  k ) #  0 )  ->  (
( A  e.  CC  /\  A #  0 )  -> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  (
k  +  1 ) ) #  0 ) ) )
563, 6, 9, 12, 30, 55nnind 8122 . . 3  |-  ( N  e.  NN  ->  (
( A  e.  CC  /\  A #  0 )  -> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) #  0 ) )
5756impcom 123 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 N ) #  0 )
58573impa 1134 1  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) #  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {csn 3406   class class class wbr 3793    X. cxp 4369   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   # cap 7748   NNcn 8106   ZZ>=cuz 8700    seqcseq 9521
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-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-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-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-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522
This theorem is referenced by:  expival  9575
  Copyright terms: Public domain W3C validator