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Theorem ntrivcvgtail 25181
Description: A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgtail.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvgtail.2  |-  ( ph  ->  N  e.  Z )
ntrivcvgtail.3  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  X )
ntrivcvgtail.4  |-  ( ph  ->  X  =/=  0 )
ntrivcvgtail.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
Assertion
Ref Expression
ntrivcvgtail  |-  ( ph  ->  ( (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0  /\  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) ) )
Distinct variable groups:    k, F    ph, k    k, M    k, N    k, Z
Allowed substitution hint:    X( k)

Proof of Theorem ntrivcvgtail
StepHypRef Expression
1 fclim 12302 . . . . . . . 8  |-  ~~>  : dom  ~~>  --> CC
2 ffun 5552 . . . . . . . 8  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
31, 2ax-mp 8 . . . . . . 7  |-  Fun  ~~>
4 ntrivcvgtail.3 . . . . . . 7  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  X )
5 funbrfv 5724 . . . . . . 7  |-  ( Fun  ~~>  ->  (  seq  M (  x.  ,  F )  ~~>  X  ->  (  ~~>  `  seq  M (  x.  ,  F
) )  =  X ) )
63, 4, 5mpsyl 61 . . . . . 6  |-  ( ph  ->  (  ~~>  `  seq  M (  x.  ,  F ) )  =  X )
7 ntrivcvgtail.4 . . . . . 6  |-  ( ph  ->  X  =/=  0 )
86, 7eqnetrd 2585 . . . . 5  |-  ( ph  ->  (  ~~>  `  seq  M (  x.  ,  F ) )  =/=  0 )
94, 6breqtrrd 4198 . . . . 5  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  (  ~~>  `  seq  M (  x.  ,  F
) ) )
108, 9jca 519 . . . 4  |-  ( ph  ->  ( (  ~~>  `  seq  M (  x.  ,  F
) )  =/=  0  /\  seq  M (  x.  ,  F )  ~~>  (  ~~>  `  seq  M (  x.  ,  F
) ) ) )
1110adantr 452 . . 3  |-  ( (
ph  /\  N  =  M )  ->  (
(  ~~>  `  seq  M (  x.  ,  F ) )  =/=  0  /\ 
seq  M (  x.  ,  F )  ~~>  (  ~~>  `  seq  M (  x.  ,  F
) ) ) )
12 seqeq1 11281 . . . . . . 7  |-  ( N  =  M  ->  seq  N (  x.  ,  F
)  =  seq  M
(  x.  ,  F
) )
1312fveq2d 5691 . . . . . 6  |-  ( N  =  M  ->  (  ~~>  ` 
seq  N (  x.  ,  F ) )  =  (  ~~>  `  seq  M (  x.  ,  F
) ) )
1413neeq1d 2580 . . . . 5  |-  ( N  =  M  ->  (
(  ~~>  `  seq  N (  x.  ,  F ) )  =/=  0  <->  (  ~~>  ` 
seq  M (  x.  ,  F ) )  =/=  0 ) )
1512, 13breq12d 4185 . . . . 5  |-  ( N  =  M  ->  (  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) )  <->  seq  M (  x.  ,  F )  ~~>  (  ~~>  `  seq  M (  x.  ,  F ) ) ) )
1614, 15anbi12d 692 . . . 4  |-  ( N  =  M  ->  (
( (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0  /\  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) )  <->  ( (  ~~>  ` 
seq  M (  x.  ,  F ) )  =/=  0  /\  seq  M (  x.  ,  F
)  ~~>  (  ~~>  `  seq  M (  x.  ,  F
) ) ) ) )
1716adantl 453 . . 3  |-  ( (
ph  /\  N  =  M )  ->  (
( (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0  /\  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) )  <->  ( (  ~~>  ` 
seq  M (  x.  ,  F ) )  =/=  0  /\  seq  M (  x.  ,  F
)  ~~>  (  ~~>  `  seq  M (  x.  ,  F
) ) ) ) )
1811, 17mpbird 224 . 2  |-  ( (
ph  /\  N  =  M )  ->  (
(  ~~>  `  seq  N (  x.  ,  F ) )  =/=  0  /\ 
seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) ) )
19 ntrivcvgtail.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
20 simpr 448 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
2120, 19syl6eleqr 2495 . . . . . 6  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( N  -  1 )  e.  Z )
22 ntrivcvgtail.5 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2322adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
244adantr 452 . . . . . 6  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  seq  M (  x.  ,  F )  ~~>  X )
257adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  X  =/=  0 )
2619, 21, 24, 25, 23ntrivcvgfvn0 25180 . . . . . 6  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  x.  ,  F
) `  ( N  -  1 ) )  =/=  0 )
2719, 21, 23, 24, 26clim2div 25170 . . . . 5  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F )  ~~>  ( X  /  (  seq  M (  x.  ,  F ) `  ( N  -  1 ) ) ) )
28 funbrfv 5724 . . . . 5  |-  ( Fun  ~~>  ->  (  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F )  ~~>  ( X  /  (  seq  M (  x.  ,  F ) `  ( N  -  1 ) ) )  ->  (  ~~>  ` 
seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F ) )  =  ( X  / 
(  seq  M (  x.  ,  F ) `  ( N  -  1
) ) ) ) )
293, 27, 28mpsyl 61 . . . 4  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  ~~>  `  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F ) )  =  ( X  /  (  seq  M (  x.  ,  F ) `  ( N  -  1 ) ) ) )
30 climcl 12248 . . . . . . 7  |-  (  seq 
M (  x.  ,  F )  ~~>  X  ->  X  e.  CC )
314, 30syl 16 . . . . . 6  |-  ( ph  ->  X  e.  CC )
3231adantr 452 . . . . 5  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  X  e.  CC )
33 ntrivcvgtail.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  Z )
34 eluzel2 10449 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3534, 19eleq2s 2496 . . . . . . . . 9  |-  ( N  e.  Z  ->  M  e.  ZZ )
3633, 35syl 16 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
3719, 36, 22prodf 25168 . . . . . . 7  |-  ( ph  ->  seq  M (  x.  ,  F ) : Z --> CC )
3819feq2i 5545 . . . . . . 7  |-  (  seq 
M (  x.  ,  F ) : Z --> CC 
<->  seq  M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3937, 38sylib 189 . . . . . 6  |-  ( ph  ->  seq  M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
4039ffvelrnda 5829 . . . . 5  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  x.  ,  F
) `  ( N  -  1 ) )  e.  CC )
4132, 40, 25, 26divne0d 9762 . . . 4  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( X  /  (  seq  M (  x.  ,  F ) `
 ( N  - 
1 ) ) )  =/=  0 )
4229, 41eqnetrd 2585 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  ~~>  `  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F ) )  =/=  0 )
4327, 29breqtrrd 4198 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F )  ~~>  (  ~~>  `  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F ) ) )
44 uzssz 10461 . . . . . . . . . . . 12  |-  ( ZZ>= `  M )  C_  ZZ
4519, 44eqsstri 3338 . . . . . . . . . . 11  |-  Z  C_  ZZ
4645, 33sseldi 3306 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
4746zcnd 10332 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
4847adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  N  e.  CC )
49 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
5049a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  1  e.  CC )
5148, 50npcand 9371 . . . . . . 7  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( ( N  -  1 )  +  1 )  =  N )
5251seqeq1d 11284 . . . . . 6  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F )  =  seq  N (  x.  ,  F ) )
5352fveq2d 5691 . . . . 5  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  ~~>  `  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F ) )  =  (  ~~>  `  seq  N (  x.  ,  F ) ) )
5453neeq1d 2580 . . . 4  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( (  ~~>  ` 
seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F ) )  =/=  0  <->  (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0
) )
5552, 53breq12d 4185 . . . 4  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  (  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F )  ~~>  (  ~~>  `  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F ) )  <->  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F ) ) ) )
5654, 55anbi12d 692 . . 3  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( (
(  ~~>  `  seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F ) )  =/=  0  /\ 
seq  ( ( N  -  1 )  +  1 ) (  x.  ,  F )  ~~>  (  ~~>  `  seq  ( ( N  - 
1 )  +  1 ) (  x.  ,  F ) ) )  <-> 
( (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0  /\  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) ) ) )
5742, 43, 56mpbi2and 888 . 2  |-  ( (
ph  /\  ( N  -  1 )  e.  ( ZZ>= `  M )
)  ->  ( (  ~~>  ` 
seq  N (  x.  ,  F ) )  =/=  0  /\  seq  N (  x.  ,  F
)  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) ) )
5833, 19syl6eleq 2494 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
59 uzm1 10472 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
6058, 59syl 16 . 2  |-  ( ph  ->  ( N  =  M  \/  ( N  - 
1 )  e.  (
ZZ>= `  M ) ) )
6118, 57, 60mpjaodan 762 1  |-  ( ph  ->  ( (  ~~>  `  seq  N (  x.  ,  F
) )  =/=  0  /\  seq  N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   dom cdm 4837   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633   ZZcz 10238   ZZ>=cuz 10444    seq cseq 11278    ~~> cli 12233
This theorem is referenced by:  ntrivcvgmullem  25182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
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