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Theorem ntrivcvg 27381
Description: A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvg.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvg.2  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
ntrivcvg.3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
Assertion
Ref Expression
ntrivcvg  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
Distinct variable groups:    k, F, n, y    ph, k, y   
k, M, n, y    ph, n, y    k, Z, y
Allowed substitution hint:    Z( n)

Proof of Theorem ntrivcvg
StepHypRef Expression
1 ntrivcvg.2 . 2  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
2 uzm1 10883 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  =  M  \/  (
n  -  1 )  e.  ( ZZ>= `  M
) ) )
3 ntrivcvg.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2530 . . . . . . . 8  |-  ( n  e.  Z  ->  (
n  =  M  \/  ( n  -  1
)  e.  ( ZZ>= `  M ) ) )
54ad2antlr 726 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  ( n  =  M  \/  ( n  -  1 )  e.  ( ZZ>= `  M )
) )
6 seqeq1 11801 . . . . . . . . . . 11  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
76breq1d 4297 . . . . . . . . . 10  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
8 seqex 11800 . . . . . . . . . . 11  |-  seq M
(  x.  ,  F
)  e.  _V
9 vex 2970 . . . . . . . . . . 11  |-  y  e. 
_V
108, 9breldm 5039 . . . . . . . . . 10  |-  (  seq M (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
117, 10syl6bi 228 . . . . . . . . 9  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
1211adantld 467 . . . . . . . 8  |-  ( n  =  M  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
13 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (
n  -  1 )  e.  Z )
14 ntrivcvg.3 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1514adantlr 714 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1615adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  k  e.  Z
)  ->  ( F `  k )  e.  CC )
1716adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  - 
1 )  e.  Z
)  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  k  e.  Z )  ->  ( F `  k
)  e.  CC )
18 uzssz 10872 . . . . . . . . . . . . . . . . . . . 20  |-  ( ZZ>= `  M )  C_  ZZ
193, 18eqsstri 3381 . . . . . . . . . . . . . . . . . . 19  |-  Z  C_  ZZ
20 simplr 754 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  Z )
2119, 20sseldi 3349 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  ZZ )
2221zcnd 10740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  CC )
23 ax-1cn 9332 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
2423a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
1  e.  CC )
2522, 24npcand 9715 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
( ( n  - 
1 )  +  1 )  =  n )
2625seqeq1d 11804 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  seq ( ( n  - 
1 )  +  1 ) (  x.  ,  F )  =  seq n (  x.  ,  F ) )
2726breq1d 4297 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
(  seq ( ( n  -  1 )  +  1 ) (  x.  ,  F )  ~~>  y  <->  seq n
(  x.  ,  F
)  ~~>  y ) )
2827biimpar 485 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq ( ( n  - 
1 )  +  1 ) (  x.  ,  F )  ~~>  y )
293, 13, 17, 28clim2prod 27372 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  ~~>  ( (  seq M (  x.  ,  F ) `  ( n  -  1
) )  x.  y
) )
30 ovex 6111 . . . . . . . . . . . . 13  |-  ( (  seq M (  x.  ,  F ) `  ( n  -  1
) )  x.  y
)  e.  _V
318, 30breldm 5039 . . . . . . . . . . . 12  |-  (  seq M (  x.  ,  F )  ~~>  ( (  seq M (  x.  ,  F ) `  ( n  -  1
) )  x.  y
)  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3229, 31syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3332an32s 802 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( n  - 
1 )  e.  Z
)  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3433expcom 435 . . . . . . . . 9  |-  ( ( n  -  1 )  e.  Z  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
353eqcomi 2442 . . . . . . . . 9  |-  ( ZZ>= `  M )  =  Z
3634, 35eleq2s 2530 . . . . . . . 8  |-  ( ( n  -  1 )  e.  ( ZZ>= `  M
)  ->  ( (
( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
3712, 36jaoi 379 . . . . . . 7  |-  ( ( n  =  M  \/  ( n  -  1
)  e.  ( ZZ>= `  M ) )  -> 
( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
385, 37mpcom 36 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
3938ex 434 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4039adantld 467 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
4140exlimdv 1690 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4241rexlimdva 2836 . 2  |-  ( ph  ->  ( E. n  e.  Z  E. y ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
431, 42mpd 15 1  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   E.wrex 2711   class class class wbr 4287   dom cdm 4835   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587   ZZcz 10638   ZZ>=cuz 10853    seqcseq 11798    ~~> cli 12954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958
This theorem is referenced by:  iprodclim2  27468  iprodcl  27470
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