Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrivcvgfvn0 Structured version   Unicode version

Theorem ntrivcvgfvn0 27365
Description: Any value of a product sequence that converges to a non-zero value is itself non-zero. (Contributed by Scott Fenton, 20-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgfvn0.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvgfvn0.2  |-  ( ph  ->  N  e.  Z )
ntrivcvgfvn0.3  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
ntrivcvgfvn0.4  |-  ( ph  ->  X  =/=  0 )
ntrivcvgfvn0.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
Assertion
Ref Expression
ntrivcvgfvn0  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  =/=  0 )
Distinct variable groups:    k, F    ph, k    k, M    k, N    k, Z
Allowed substitution hint:    X( k)

Proof of Theorem ntrivcvgfvn0
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ntrivcvgfvn0.4 . 2  |-  ( ph  ->  X  =/=  0 )
2 fclim 13023 . . . . . . . 8  |-  ~~>  : dom  ~~>  --> CC
3 ffun 5556 . . . . . . . 8  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
42, 3ax-mp 5 . . . . . . 7  |-  Fun  ~~>
5 ntrivcvgfvn0.3 . . . . . . 7  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
6 funbrfv 5725 . . . . . . 7  |-  ( Fun  ~~>  ->  (  seq M (  x.  ,  F )  ~~>  X  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X ) )
74, 5, 6mpsyl 63 . . . . . 6  |-  ( ph  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X )
87adantr 465 . . . . 5  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X )
9 eqid 2438 . . . . . . 7  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
10 ntrivcvgfvn0.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
11 uzssz 10872 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
1210, 11eqsstri 3381 . . . . . . . . 9  |-  Z  C_  ZZ
13 ntrivcvgfvn0.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
1412, 13sseldi 3349 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  N  e.  ZZ )
16 seqex 11800 . . . . . . . 8  |-  seq M
(  x.  ,  F
)  e.  _V
1716a1i 11 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  e. 
_V )
18 0cnd 9371 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  0  e.  CC )
19 fveq2 5686 . . . . . . . . . . 11  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2019eqeq1d 2446 . . . . . . . . . 10  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  N )  =  0 ) )
2120imbi2d 316 . . . . . . . . 9  |-  ( m  =  N  ->  (
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  0 )  <-> 
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  N
)  =  0 ) ) )
22 fveq2 5686 . . . . . . . . . . 11  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
2322eqeq1d 2446 . . . . . . . . . 10  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  n )  =  0 ) )
2423imbi2d 316 . . . . . . . . 9  |-  ( m  =  n  ->  (
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  0 )  <-> 
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  n
)  =  0 ) ) )
25 fveq2 5686 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
2625eqeq1d 2446 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  ( n  +  1
) )  =  0 ) )
2726imbi2d 316 . . . . . . . . 9  |-  ( m  =  ( n  + 
1 )  ->  (
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  0 )  <-> 
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 ) ) )
28 fveq2 5686 . . . . . . . . . . 11  |-  ( m  =  k  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  k
) )
2928eqeq1d 2446 . . . . . . . . . 10  |-  ( m  =  k  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  k )  =  0 ) )
3029imbi2d 316 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  0 )  <-> 
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  k
)  =  0 ) ) )
31 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  seq M
(  x.  ,  F
) `  N )  =  0 )
3231a1i 11 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  seq M
(  x.  ,  F
) `  N )  =  0 ) )
3313, 10syl6eleq 2528 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
34 uztrn 10869 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( ZZ>= `  N )  /\  N  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
3533, 34sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  n  e.  (
ZZ>= `  M ) )
36353adant3 1008 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  n  e.  ( ZZ>= `  M )
)
37 seqp1 11813 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
3836, 37syl 16 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
39 oveq1 6093 . . . . . . . . . . . . . 14  |-  ( (  seq M (  x.  ,  F ) `  n )  =  0  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =  ( 0  x.  ( F `  (
n  +  1 ) ) ) )
40393ad2ant3 1011 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (
(  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) )  =  ( 0  x.  ( F `  ( n  +  1
) ) ) )
41 peano2uz 10900 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( n  +  1 )  e.  ( ZZ>= `  N )
)
4210uztrn2 10870 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  Z  /\  ( n  +  1
)  e.  ( ZZ>= `  N ) )  -> 
( n  +  1 )  e.  Z )
4313, 41, 42syl2an 477 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  ( n  +  1 )  e.  Z )
44 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
4544ralrimiva 2794 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
46 fveq2 5686 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4746eleq1d 2504 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
4847rspcv 3064 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  ( n  +  1 ) )  e.  CC ) )
4945, 48mpan9 469 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  ( F `  ( n  +  1 ) )  e.  CC )
5043, 49syldan 470 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
5150ancoms 453 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  ( F `  ( n  +  1
) )  e.  CC )
5251mul02d 9559 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  ( 0  x.  ( F `  (
n  +  1 ) ) )  =  0 )
53523adant3 1008 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (
0  x.  ( F `
 ( n  + 
1 ) ) )  =  0 )
5438, 40, 533eqtrd 2474 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 )
55543exp 1186 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  n )  =  0  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 ) ) )
5655adantrd 468 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( ( ph  /\  (  seq M
(  x.  ,  F
) `  N )  =  0 )  -> 
( (  seq M
(  x.  ,  F
) `  n )  =  0  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 ) ) )
5756a2d 26 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( (
( ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  seq M
(  x.  ,  F
) `  n )  =  0 )  -> 
( ( ph  /\  (  seq M (  x.  ,  F ) `  N )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 ) ) )
5821, 24, 27, 30, 32, 57uzind4 10904 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( ( ph  /\  (  seq M
(  x.  ,  F
) `  N )  =  0 )  -> 
(  seq M (  x.  ,  F ) `  k )  =  0 ) )
5958impcom 430 . . . . . . 7  |-  ( ( ( ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  /\  k  e.  (
ZZ>= `  N ) )  ->  (  seq M
(  x.  ,  F
) `  k )  =  0 )
609, 15, 17, 18, 59climconst 13013 . . . . . 6  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  ~~>  0 )
61 funbrfv 5725 . . . . . 6  |-  ( Fun  ~~>  ->  (  seq M (  x.  ,  F )  ~~>  0  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  0 ) )
624, 60, 61mpsyl 63 . . . . 5  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  0 )
638, 62eqtr3d 2472 . . . 4  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  X  =  0 )
6463ex 434 . . 3  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  =  0  ->  X  =  0 ) )
6564necon3d 2641 . 2  |-  ( ph  ->  ( X  =/=  0  ->  (  seq M (  x.  ,  F ) `
 N )  =/=  0 ) )
661, 65mpd 15 1  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967   class class class wbr 4287   dom cdm 4835   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   ZZcz 10638   ZZ>=cuz 10853    seqcseq 11798    ~~> cli 12954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958
This theorem is referenced by:  ntrivcvgtail  27366
  Copyright terms: Public domain W3C validator