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Theorem ntrivcvgfvn0 13942
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 13604 . . . . . . . 8  |-  ~~>  : dom  ~~>  --> CC
3 ffun 5744 . . . . . . . 8  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
42, 3ax-mp 5 . . . . . . 7  |-  Fun  ~~>
5 ntrivcvgfvn0.3 . . . . . . 7  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
6 funbrfv 5915 . . . . . . 7  |-  ( Fun  ~~>  ->  (  seq M (  x.  ,  F )  ~~>  X  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X ) )
74, 5, 6mpsyl 65 . . . . . 6  |-  ( ph  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X )
87adantr 466 . . . . 5  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  X )
9 eqid 2422 . . . . . . 7  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
10 ntrivcvgfvn0.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
11 uzssz 11178 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
1210, 11eqsstri 3494 . . . . . . . . 9  |-  Z  C_  ZZ
13 ntrivcvgfvn0.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
1412, 13sseldi 3462 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1514adantr 466 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  N  e.  ZZ )
16 seqex 12214 . . . . . . . 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 9636 . . . . . . 7  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  0  e.  CC )
19 fveq2 5877 . . . . . . . . . . 11  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2019eqeq1d 2424 . . . . . . . . . 10  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  N )  =  0 ) )
2120imbi2d 317 . . . . . . . . 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 5877 . . . . . . . . . . 11  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
2322eqeq1d 2424 . . . . . . . . . 10  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  n )  =  0 ) )
2423imbi2d 317 . . . . . . . . 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 5877 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
2625eqeq1d 2424 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  ( n  +  1
) )  =  0 ) )
2726imbi2d 317 . . . . . . . . 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 5877 . . . . . . . . . . 11  |-  ( m  =  k  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  k
) )
2928eqeq1d 2424 . . . . . . . . . 10  |-  ( m  =  k  ->  (
(  seq M (  x.  ,  F ) `  m )  =  0  <-> 
(  seq M (  x.  ,  F ) `  k )  =  0 ) )
3029imbi2d 317 . . . . . . . . 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 462 . . . . . . . . . 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 2520 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
34 uztrn 11175 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( ZZ>= `  N )  /\  N  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
3533, 34sylan2 476 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  n  e.  (
ZZ>= `  M ) )
36353adant3 1025 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  n  e.  ( ZZ>= `  M )
)
37 seqp1 12227 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
3836, 37syl 17 . . . . . . . . . . . . 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 6308 . . . . . . . . . . . . . 14  |-  ( (  seq M (  x.  ,  F ) `  n )  =  0  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =  ( 0  x.  ( F `  (
n  +  1 ) ) ) )
40393ad2ant3 1028 . . . . . . . . . . . . 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 11212 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( n  +  1 )  e.  ( ZZ>= `  N )
)
4210uztrn2 11176 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  Z  /\  ( n  +  1
)  e.  ( ZZ>= `  N ) )  -> 
( n  +  1 )  e.  Z )
4313, 41, 42syl2an 479 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  ( n  +  1 )  e.  Z )
44 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
4544ralrimiva 2839 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
46 fveq2 5877 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4746eleq1d 2491 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
4847rspcv 3178 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  ( n  +  1 ) )  e.  CC ) )
4945, 48mpan9 471 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  ( F `  ( n  +  1 ) )  e.  CC )
5043, 49syldan 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( ZZ>= `  N )
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
5150ancoms 454 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  ( F `  ( n  +  1
) )  e.  CC )
5251mul02d 9831 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph )  ->  ( 0  x.  ( F `  (
n  +  1 ) ) )  =  0 )
53523adant3 1025 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (
0  x.  ( F `
 ( n  + 
1 ) ) )  =  0 )
5438, 40, 533eqtrd 2467 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ZZ>= `  N )  /\  ph  /\  (  seq M (  x.  ,  F ) `
 n )  =  0 )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 )
55543exp 1204 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  N
)  ->  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  n )  =  0  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  0 ) ) )
5655adantrd 469 . . . . . . . . . 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 29 . . . . . . . . 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 11217 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( ( ph  /\  (  seq M
(  x.  ,  F
) `  N )  =  0 )  -> 
(  seq M (  x.  ,  F ) `  k )  =  0 ) )
5958impcom 431 . . . . . . 7  |-  ( ( ( ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  /\  k  e.  (
ZZ>= `  N ) )  ->  (  seq M
(  x.  ,  F
) `  k )  =  0 )
609, 15, 17, 18, 59climconst 13594 . . . . . 6  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  seq M (  x.  ,  F )  ~~>  0 )
61 funbrfv 5915 . . . . . 6  |-  ( Fun  ~~>  ->  (  seq M (  x.  ,  F )  ~~>  0  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  0 ) )
624, 60, 61mpsyl 65 . . . . 5  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  (  ~~>  `  seq M (  x.  ,  F ) )  =  0 )
638, 62eqtr3d 2465 . . . 4  |-  ( (
ph  /\  (  seq M (  x.  ,  F ) `  N
)  =  0 )  ->  X  =  0 )
6463ex 435 . . 3  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  =  0  ->  X  =  0 ) )
6564necon3d 2648 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   _Vcvv 3081   class class class wbr 4420   dom cdm 4849   Fun wfun 5591   -->wf 5593   ` cfv 5597  (class class class)co 6301   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   ZZcz 10937   ZZ>=cuz 11159    seqcseq 12212    ~~> cli 13535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539
This theorem is referenced by:  ntrivcvgtail  13943
  Copyright terms: Public domain W3C validator