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Theorem exp3vallem 10464
Description: Lemma for exp3val 10465. 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 5494 . . . . 5  |-  ( w  =  1  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 1 ) )
32breq1d 3997 . . . 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 5494 . . . . 5  |-  ( w  =  k  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 k ) )
65breq1d 3997 . . . 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 5494 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
98breq1d 3997 . . . 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 5494 . . . . 5  |-  ( w  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  w )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
1211breq1d 3997 . . . 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 9226 . . . . . 6  |-  ( ph  ->  1  e.  ZZ )
15 exp3vallem.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
16 elnnuz 9510 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
1716biimpri 132 . . . . . . . 8  |-  ( x  e.  ( ZZ>= `  1
)  ->  x  e.  NN )
18 fvconst2g 5707 . . . . . . . 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 2247 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  x )  e.  CC )
22 mulcl 7888 . . . . . . 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 10403 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  ( ( NN  X.  { A } ) `  1
) )
25 1nn 8876 . . . . . 6  |-  1  e.  NN
26 fvconst2g 5707 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2715, 25, 26sylancl 411 . . . . 5  |-  ( ph  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2824, 27eqtrd 2203 . . . 4  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
)  =  A )
29 exp3vallem.ap . . . 4  |-  ( ph  ->  A #  0 )
3028, 29eqbrtrd 4009 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  1
) #  0 )
31 nnuz 9509 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3216, 21sylan2b 285 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x
)  e.  CC )
3331, 14, 32, 23seqf 10404 . . . . . . . . . 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 5629 . . . . . . . 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 486 . . . . . . 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 486 . . . . . . 7  |-  ( ( ( k  e.  NN  /\ 
ph )  /\  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  k ) #  0 )  ->  A #  0 )
4137, 38, 39, 40mulap0d 8563 . . . . . 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 9510 . . . . . . . . . . . 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 473 . . . . . . . . . 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 10405 . . . . . . . . 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 8880 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ph )  ->  ( k  +  1 )  e.  NN )
49 fvconst2g 5707 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
5015, 48, 49syl2an2 589 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  ph )  ->  ( ( NN  X.  { A }
) `  ( k  +  1 ) )  =  A )
5150oveq2d 5866 . . . . . . . . 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 2203 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ph )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  k
)  x.  A ) )
5352breq1d 3997 . . . . . . 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 362 . . . 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 8881 . 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 1348    e. wcel 2141   {csn 3581   class class class wbr 3987    X. cxp 4607   -->wf 5192   ` cfv 5196  (class class class)co 5850   CCcc 7759   0cc0 7761   1c1 7762    + caddc 7764    x. cmul 7766   # cap 8487   NNcn 8865   ZZ>=cuz 9474    seqcseq 10388
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-seqfrec 10389
This theorem is referenced by:  exp3val  10465
  Copyright terms: Public domain W3C validator