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Theorem ntrivcvgfvn0 28960
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 13356 . . . . . . . 8  |-  ~~>  : dom  ~~>  --> CC
3 ffun 5739 . . . . . . . 8  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
42, 3ax-mp 5 . . . . . . 7  |-  Fun  ~~>
5 ntrivcvgfvn0.3 . . . . . . 7  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
6 funbrfv 5912 . . . . . . 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 2467 . . . . . . 7  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
10 ntrivcvgfvn0.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
11 uzssz 11113 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
1210, 11eqsstri 3539 . . . . . . . . 9  |-  Z  C_  ZZ
13 ntrivcvgfvn0.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
1412, 13sseldi 3507 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  N  e.  ZZ )
16 seqex 12089 . . . . . . . 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 9601 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  0  e.  CC )
19 fveq2 5872 . . . . . . . . . . 11  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2019eqeq1d 2469 . . . . . . . . . 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 5872 . . . . . . . . . . 11  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
2322eqeq1d 2469 . . . . . . . . . 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 5872 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
2625eqeq1d 2469 . . . . . . . . . 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 5872 . . . . . . . . . . 11  |-  ( m  =  k  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  k
) )
2928eqeq1d 2469 . . . . . . . . . 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 2565 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
34 uztrn 11110 . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  n  e.  ( ZZ>= `  M )
)
37 seqp1 12102 . . . . . . . . . . . . . 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 6302 . . . . . . . . . . . . . 14  |-  ( (  seq M (  x.  ,  F ) `  n )  =  0  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =  ( 0  x.  ( F `  (
n  +  1 ) ) ) )
40393ad2ant3 1019 . . . . . . . . . . . . 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 11146 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( n  +  1 )  e.  ( ZZ>= `  N )
)
4210uztrn2 11111 . . . . . . . . . . . . . . . . . 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 2881 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
46 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4746eleq1d 2536 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
4847rspcv 3215 . . . . . . . . . . . . . . . . . 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 9789 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  ( 0  x.  ( F `  (
n  +  1 ) ) )  =  0 )
53523adant3 1016 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (
0  x.  ( F `
 ( n  + 
1 ) ) )  =  0 )
5438, 40, 533eqtrd 2512 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 )
55543exp 1195 . . . . . . . . . . 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 11151 . . . . . . . 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 13346 . . . . . 6  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  ~~>  0 )
61 funbrfv 5912 . . . . . 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 2510 . . . 4  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  X  =  0 )
6463ex 434 . . 3  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  =  0  ->  X  =  0 ) )
6564necon3d 2691 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118   class class class wbr 4453   dom cdm 5005   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   ZZcz 10876   ZZ>=cuz 11094    seqcseq 12087    ~~> cli 13287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291
This theorem is referenced by:  ntrivcvgtail  28961
  Copyright terms: Public domain W3C validator