ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exp3vallem Unicode version

Theorem exp3vallem 10019
Description: Lemma for exp3val 10020. 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.)
Hypotheses
Ref Expression
exp3vallem.a  |-  ( ph  ->  A  e.  CC )
exp3vallem.ap  |-  ( ph  ->  A #  0 )
exp3vallem.n  |-  ( ph  ->  N  e.  NN )
Assertion
Ref Expression
exp3vallem  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) #  0 )

Proof of Theorem exp3vallem
Dummy variables  k  x  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exp3vallem.n . 2  |-  ( ph  ->  N  e.  NN )
2 fveq2 5320 . . . . 5  |-  ( w  =  1  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) )
32breq1d 3863 . . . 4  |-  ( w  =  1  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) #  0 ) )
43imbi2d 229 . . 3  |-  ( w  =  1  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) ` 
1 ) #  0 ) ) )
5 fveq2 5320 . . . . 5  |-  ( w  =  k  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) )
65breq1d 3863 . . . 4  |-  ( w  =  k  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) #  0 ) )
76imbi2d 229 . . 3  |-  ( w  =  k  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 ) ) )
8 fveq2 5320 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
98breq1d 3863 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) #  0 ) )
109imbi2d 229 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) ) #  0 ) ) )
11 fveq2 5320 . . . . 5  |-  ( w  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
1211breq1d 3863 . . . 4  |-  ( w  =  N  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) #  0 ) )
1312imbi2d 229 . . 3  |-  ( w  =  N  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) #  0 ) ) )
14 1zzd 8840 . . . . . 6  |-  ( ph  ->  1  e.  ZZ )
15 exp3vallem.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
16 elnnuz 9118 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
1716biimpri 132 . . . . . . . 8  |-  ( x  e.  ( ZZ>= `  1
)  ->  x  e.  NN )
18 fvconst2g 5527 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
1915, 17, 18syl2an 284 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  =  A )
2015adantr 271 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  A  e.  CC )
2119, 20eqeltrd 2165 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
22 mulcl 7532 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 272 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
2414, 21, 23seq3-1 9940 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  ( ( NN  X.  { A } ) `  1
) )
25 1nn 8496 . . . . . 6  |-  1  e.  NN
26 fvconst2g 5527 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2715, 25, 26sylancl 405 . . . . 5  |-  ( ph  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2824, 27eqtrd 2121 . . . 4  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  A )
29 exp3vallem.ap . . . 4  |-  ( ph  ->  A #  0 )
3028, 29eqbrtrd 3873 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
) #  0 )
31 nnuz 9117 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3216, 21sylan2b 282 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x
)  e.  CC )
3331, 14, 32, 23seqf 9943 . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
3433adantl 272 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  seq 1
(  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
35 simpl 108 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  k  e.  NN )
3634, 35ffvelrnd 5451 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k )  e.  CC )
3736adantr 271 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  k )  e.  CC )
3815ad2antlr 474 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A  e.  CC )
39 simpr 109 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  k ) #  0 )
4029ad2antlr 474 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A #  0 )
4137, 38, 39, 40mulap0d 8190 . . . . . 6  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k )  x.  A
) #  0 )
42 elnnuz 9118 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
4342biimpi 119 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  ( ZZ>= `  1 )
)
4443adantr 271 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  k  e.  ( ZZ>= `  1 )
)
4521adantll 461 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
4622adantl 272 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  e.  CC )
4744, 45, 46seq3p1 9947 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) ) )
4835peano2nnd 8500 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ph )  ->  ( k  +  1 )  e.  NN )
49 fvconst2g 5527 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
5015, 48, 49syl2an2 562 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  ( ( NN  X.  { A }
) `  ( k  +  1 ) )  =  A )
5150oveq2d 5684 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k )  x.  (
( NN  X.  { A } ) `  (
k  +  1 ) ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  A ) )
5247, 51eqtrd 2121 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  A ) )
5352breq1d 3863 . . . . . . 7  |-  ( ( k  e.  NN  /\  ph )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) ) #  0  <->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k )  x.  A ) #  0 ) )
5453adantr 271 . . . . . 6  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) ) #  0  <->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k )  x.  A ) #  0 ) )
5541, 54mpbird 166 . . . . 5  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  ( k  +  1 ) ) #  0 )
5655exp31 357 . . . 4  |-  ( k  e.  NN  ->  ( ph  ->  ( (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0  ->  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) #  0 ) ) )
5756a2d 26 . . 3  |-  ( k  e.  NN  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  ( ph  ->  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) #  0 ) ) )
584, 7, 10, 13, 30, 57nnind 8501 . 2  |-  ( N  e.  NN  ->  ( ph  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  N ) #  0 ) )
591, 58mpcom 36 1  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) #  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   {csn 3452   class class class wbr 3853    X. cxp 4452   -->wf 5026   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414    + caddc 7416    x. cmul 7418   # cap 8121   NNcn 8485   ZZ>=cuz 9082    seqcseq 9915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-frec 6172  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-inn 8486  df-n0 8737  df-z 8814  df-uz 9083  df-iseq 9916  df-seq3 9917
This theorem is referenced by:  exp3val  10020
  Copyright terms: Public domain W3C validator