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Theorem ntrivcvgfvn0 13720
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 13388 . . . . . . . 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 5911 . . . . . . 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 2457 . . . . . . 7  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
10 ntrivcvgfvn0.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
11 uzssz 11125 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
1210, 11eqsstri 3529 . . . . . . . . 9  |-  Z  C_  ZZ
13 ntrivcvgfvn0.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
1412, 13sseldi 3497 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  N  e.  ZZ )
16 seqex 12112 . . . . . . . 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 9606 . . . . . . 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 2459 . . . . . . . . . 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 2459 . . . . . . . . . 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 2459 . . . . . . . . . 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 2459 . . . . . . . . . 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 2555 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
34 uztrn 11122 . . . . . . . . . . . . . . . 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 12125 . . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . . 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 11159 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( n  +  1 )  e.  ( ZZ>= `  N )
)
4210uztrn2 11123 . . . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
46 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4746eleq1d 2526 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
4847rspcv 3206 . . . . . . . . . . . . . . . . . 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 9795 . . . . . . . . . . . . . 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 2502 . . . . . . . . . . . 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 11164 . . . . . . . 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 13378 . . . . . 6  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  ~~>  0 )
61 funbrfv 5911 . . . . . 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 2500 . . . 4  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  X  =  0 )
6463ex 434 . . 3  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  =  0  ->  X  =  0 ) )
6564necon3d 2681 . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109   class class class wbr 4456   dom cdm 5008   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   ZZcz 10885   ZZ>=cuz 11106    seqcseq 12110    ~~> cli 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323
This theorem is referenced by:  ntrivcvgtail  13721
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