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

Theorem exp3vallem 10552
Description: Lemma for exp3val 10553. 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 5534 . . . . 5  |-  ( w  =  1  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) )
32breq1d 4028 . . . 4  |-  ( w  =  1  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) #  0 ) )
43imbi2d 230 . . 3  |-  ( w  =  1  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) ` 
1 ) #  0 ) ) )
5 fveq2 5534 . . . . 5  |-  ( w  =  k  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) )
65breq1d 4028 . . . 4  |-  ( w  =  k  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) #  0 ) )
76imbi2d 230 . . 3  |-  ( w  =  k  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 ) ) )
8 fveq2 5534 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
98breq1d 4028 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) #  0 ) )
109imbi2d 230 . . 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 5534 . . . . 5  |-  ( w  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
1211breq1d 4028 . . . 4  |-  ( w  =  N  ->  (
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 w ) #  0  <-> 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) #  0 ) )
1312imbi2d 230 . . 3  |-  ( w  =  N  ->  (
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w ) #  0 )  <-> 
( ph  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) #  0 ) ) )
14 1zzd 9310 . . . . . 6  |-  ( ph  ->  1  e.  ZZ )
15 exp3vallem.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
16 elnnuz 9594 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
1716biimpri 133 . . . . . . . 8  |-  ( x  e.  ( ZZ>= `  1
)  ->  x  e.  NN )
18 fvconst2g 5751 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
1915, 17, 18syl2an 289 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  =  A )
2015adantr 276 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  A  e.  CC )
2119, 20eqeltrd 2266 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
22 mulcl 7968 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 277 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
2414, 21, 23seq3-1 10491 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  ( ( NN  X.  { A } ) `  1
) )
25 1nn 8960 . . . . . 6  |-  1  e.  NN
26 fvconst2g 5751 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2715, 25, 26sylancl 413 . . . . 5  |-  ( ph  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2824, 27eqtrd 2222 . . . 4  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  A )
29 exp3vallem.ap . . . 4  |-  ( ph  ->  A #  0 )
3028, 29eqbrtrd 4040 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
) #  0 )
31 nnuz 9593 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3216, 21sylan2b 287 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x
)  e.  CC )
3331, 14, 32, 23seqf 10492 . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
3433adantl 277 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  seq 1
(  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
35 simpl 109 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  ph )  ->  k  e.  NN )
3634, 35ffvelcdmd 5673 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k )  e.  CC )
3736adantr 276 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  k )  e.  CC )
3815ad2antlr 489 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A  e.  CC )
39 simpr 110 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  k ) #  0 )
4029ad2antlr 489 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A #  0 )
4137, 38, 39, 40mulap0d 8645 . . . . . 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 9594 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
4342biimpi 120 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  ( ZZ>= `  1 )
)
4443adantr 276 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  k  e.  ( ZZ>= `  1 )
)
4521adantll 476 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
4622adantl 277 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  e.  CC )
4744, 45, 46seq3p1 10493 . . . . . . . . 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 8964 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ph )  ->  ( k  +  1 )  e.  NN )
49 fvconst2g 5751 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
5015, 48, 49syl2an2 594 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  ( ( NN  X.  { A }
) `  ( k  +  1 ) )  =  A )
5150oveq2d 5912 . . . . . . . . 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 2222 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  A ) )
5352breq1d 4028 . . . . . . 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 276 . . . . . 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 167 . . . . 5  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  ( k  +  1 ) ) #  0 )
5655exp31 364 . . . 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 8965 . 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   {csn 3607   class class class wbr 4018    X. cxp 4642   -->wf 5231   ` cfv 5235  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842    + caddc 7844    x. cmul 7846   # cap 8568   NNcn 8949   ZZ>=cuz 9558    seqcseq 10476
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477
This theorem is referenced by:  exp3val  10553
  Copyright terms: Public domain W3C validator