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Theorem ntrivcvgfvn0 27578
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 13152 . . . . . . . 8  |-  ~~>  : dom  ~~>  --> CC
3 ffun 5672 . . . . . . . 8  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
42, 3ax-mp 5 . . . . . . 7  |-  Fun  ~~>
5 ntrivcvgfvn0.3 . . . . . . 7  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
6 funbrfv 5842 . . . . . . 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 2454 . . . . . . 7  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
10 ntrivcvgfvn0.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
11 uzssz 10994 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
1210, 11eqsstri 3497 . . . . . . . . 9  |-  Z  C_  ZZ
13 ntrivcvgfvn0.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
1412, 13sseldi 3465 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  N  e.  ZZ )
16 seqex 11928 . . . . . . . 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 9493 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  0  e.  CC )
19 fveq2 5802 . . . . . . . . . . 11  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2019eqeq1d 2456 . . . . . . . . . 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 5802 . . . . . . . . . . 11  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
2322eqeq1d 2456 . . . . . . . . . 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 5802 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
2625eqeq1d 2456 . . . . . . . . . 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 5802 . . . . . . . . . . 11  |-  ( m  =  k  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  k
) )
2928eqeq1d 2456 . . . . . . . . . 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 2552 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
34 uztrn 10991 . . . . . . . . . . . . . . . 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 11941 . . . . . . . . . . . . . 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 6210 . . . . . . . . . . . . . 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 11022 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( n  +  1 )  e.  ( ZZ>= `  N )
)
4210uztrn2 10992 . . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
46 fveq2 5802 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4746eleq1d 2523 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
4847rspcv 3175 . . . . . . . . . . . . . . . . . 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 9681 . . . . . . . . . . . . . 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 2499 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 )
55543exp 1187 . . . . . . . . . . 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 11026 . . . . . . . 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 13142 . . . . . 6  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  ~~>  0 )
61 funbrfv 5842 . . . . . 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 2497 . . . 4  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  X  =  0 )
6463ex 434 . . 3  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  =  0  ->  X  =  0 ) )
6564necon3d 2676 . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078   class class class wbr 4403   dom cdm 4951   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   ZZcz 10760   ZZ>=cuz 10975    seqcseq 11926    ~~> cli 13083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087
This theorem is referenced by:  ntrivcvgtail  27579
  Copyright terms: Public domain W3C validator