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Theorem exp3vallem 10287
Description: Lemma for exp3val 10288. 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 5414 . . . . 5  |-  ( w  =  1  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) )
32breq1d 3934 . . . 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 5414 . . . . 5  |-  ( w  =  k  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) )
65breq1d 3934 . . . 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 5414 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
98breq1d 3934 . . . 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 5414 . . . . 5  |-  ( w  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
1211breq1d 3934 . . . 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 9074 . . . . . 6  |-  ( ph  ->  1  e.  ZZ )
15 exp3vallem.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
16 elnnuz 9355 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
1716biimpri 132 . . . . . . . 8  |-  ( x  e.  ( ZZ>= `  1
)  ->  x  e.  NN )
18 fvconst2g 5627 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
1915, 17, 18syl2an 287 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  =  A )
2015adantr 274 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  A  e.  CC )
2119, 20eqeltrd 2214 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
22 mulcl 7740 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 275 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
2414, 21, 23seq3-1 10226 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  ( ( NN  X.  { A } ) `  1
) )
25 1nn 8724 . . . . . 6  |-  1  e.  NN
26 fvconst2g 5627 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2715, 25, 26sylancl 409 . . . . 5  |-  ( ph  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2824, 27eqtrd 2170 . . . 4  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  A )
29 exp3vallem.ap . . . 4  |-  ( ph  ->  A #  0 )
3028, 29eqbrtrd 3945 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
) #  0 )
31 nnuz 9354 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3216, 21sylan2b 285 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x
)  e.  CC )
3331, 14, 32, 23seqf 10227 . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
3433adantl 275 . . . . . . . . 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 5549 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k )  e.  CC )
3736adantr 274 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  k )  e.  CC )
3815ad2antlr 480 . . . . . . 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 480 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A #  0 )
4137, 38, 39, 40mulap0d 8412 . . . . . 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 9355 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
4342biimpi 119 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  ( ZZ>= `  1 )
)
4443adantr 274 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  k  e.  ( ZZ>= `  1 )
)
4521adantll 467 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
4622adantl 275 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  e.  CC )
4744, 45, 46seq3p1 10228 . . . . . . . . 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 8728 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ph )  ->  ( k  +  1 )  e.  NN )
49 fvconst2g 5627 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
5015, 48, 49syl2an2 583 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  ( ( NN  X.  { A }
) `  ( k  +  1 ) )  =  A )
5150oveq2d 5783 . . . . . . . . 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 2170 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  A ) )
5352breq1d 3934 . . . . . . 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 274 . . . . . 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 361 . . . 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 8729 . 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 1331    e. wcel 1480   {csn 3522   class class class wbr 3924    X. cxp 4532   -->wf 5114   ` cfv 5118  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618   # cap 8336   NNcn 8713   ZZ>=cuz 9319    seqcseq 10211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-seqfrec 10212
This theorem is referenced by:  exp3val  10288
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