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Theorem ntrivcvgap 11540
Description: A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvg.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvgap.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
ntrivcvgap  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
Distinct variable groups:    k, F, n, y    k, M, n, y    k, Z, y    ph, k, n, y
Allowed substitution hint:    Z( n)

Proof of Theorem ntrivcvgap
StepHypRef Expression
1 ntrivcvgap.2 . 2  |-  ( ph  ->  E. n  e.  Z  E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y ) )
2 uzm1 9547 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  =  M  \/  (
n  -  1 )  e.  ( ZZ>= `  M
) ) )
3 ntrivcvg.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2272 . . . . . . . 8  |-  ( n  e.  Z  ->  (
n  =  M  \/  ( n  -  1
)  e.  ( ZZ>= `  M ) ) )
54ad2antlr 489 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  ( n  =  M  \/  ( n  -  1 )  e.  ( ZZ>= `  M )
) )
6 seqeq1 10434 . . . . . . . . . . 11  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
76breq1d 4010 . . . . . . . . . 10  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
8 seqex 10433 . . . . . . . . . . 11  |-  seq M
(  x.  ,  F
)  e.  _V
9 vex 2740 . . . . . . . . . . 11  |-  y  e. 
_V
108, 9breldm 4827 . . . . . . . . . 10  |-  (  seq M (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
117, 10syl6bi 163 . . . . . . . . 9  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
1211adantld 278 . . . . . . . 8  |-  ( n  =  M  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
13 eluzel2 9522 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1413, 3eleq2s 2272 . . . . . . . . . . . . . . . 16  |-  ( n  e.  Z  ->  M  e.  ZZ )
1514ad3antlr 493 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  M  e.  ZZ )
16 ntrivcvg.3 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1716ad5ant15 521 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  - 
1 )  e.  Z
)  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  k  e.  Z )  ->  ( F `  k
)  e.  CC )
183, 15, 17prodf 11530 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F ) : Z --> CC )
19 simplr 528 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (
n  -  1 )  e.  Z )
2018, 19ffvelcdmd 5648 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (  seq M (  x.  ,  F ) `  (
n  -  1 ) )  e.  CC )
21 climcl 11274 . . . . . . . . . . . . . 14  |-  (  seq n (  x.  ,  F )  ~~>  y  -> 
y  e.  CC )
2221adantl 277 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  y  e.  CC )
2320, 22mulcld 7968 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (
(  seq M (  x.  ,  F ) `  ( n  -  1
) )  x.  y
)  e.  CC )
24 uzssz 9536 . . . . . . . . . . . . . . . . . . . 20  |-  ( ZZ>= `  M )  C_  ZZ
253, 24eqsstri 3187 . . . . . . . . . . . . . . . . . . 19  |-  Z  C_  ZZ
26 simplr 528 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  Z )
2725, 26sselid 3153 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  ZZ )
2827zcnd 9365 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  CC )
29 1cnd 7964 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
1  e.  CC )
3028, 29npcand 8262 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
( ( n  - 
1 )  +  1 )  =  n )
3130seqeq1d 10437 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  seq ( ( n  - 
1 )  +  1 ) (  x.  ,  F )  =  seq n (  x.  ,  F ) )
3231breq1d 4010 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
(  seq ( ( n  -  1 )  +  1 ) (  x.  ,  F )  ~~>  y  <->  seq n
(  x.  ,  F
)  ~~>  y ) )
3332biimpar 297 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq ( ( n  - 
1 )  +  1 ) (  x.  ,  F )  ~~>  y )
343, 19, 17, 33clim2prod 11531 . . . . . . . . . . . 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
) )
35 breldmg 4829 . . . . . . . . . . . 12  |-  ( (  seq M (  x.  ,  F )  e. 
_V  /\  ( (  seq M (  x.  ,  F ) `  (
n  -  1 ) )  x.  y )  e.  CC  /\  seq M (  x.  ,  F )  ~~>  ( (  seq M (  x.  ,  F ) `  ( n  -  1
) )  x.  y
) )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
368, 23, 34, 35mp3an2i 1342 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3736an32s 568 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( n  - 
1 )  e.  Z
)  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3837expcom 116 . . . . . . . . 9  |-  ( ( n  -  1 )  e.  Z  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
393eqcomi 2181 . . . . . . . . 9  |-  ( ZZ>= `  M )  =  Z
4038, 39eleq2s 2272 . . . . . . . 8  |-  ( ( n  -  1 )  e.  ( ZZ>= `  M
)  ->  ( (
( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
4112, 40jaoi 716 . . . . . . 7  |-  ( ( n  =  M  \/  ( n  -  1
)  e.  ( ZZ>= `  M ) )  -> 
( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
425, 41mpcom 36 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
4342ex 115 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4443adantld 278 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  (
( y #  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
4544exlimdv 1819 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4645rexlimdva 2594 . 2  |-  ( ph  ->  ( E. n  e.  Z  E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
471, 46mpd 13 1  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   _Vcvv 2737   class class class wbr 4000   dom cdm 4623   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    - cmin 8118   # cap 8528   ZZcz 9242   ZZ>=cuz 9517    seqcseq 10431    ~~> cli 11270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271
This theorem is referenced by: (None)
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