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Theorem ntrivcvgap 11329
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 9368 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  =  M  \/  (
n  -  1 )  e.  ( ZZ>= `  M
) ) )
3 ntrivcvg.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2234 . . . . . . . 8  |-  ( n  e.  Z  ->  (
n  =  M  \/  ( n  -  1
)  e.  ( ZZ>= `  M ) ) )
54ad2antlr 480 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  ( n  =  M  \/  ( n  -  1 )  e.  ( ZZ>= `  M )
) )
6 seqeq1 10233 . . . . . . . . . . 11  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
76breq1d 3939 . . . . . . . . . 10  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
8 seqex 10232 . . . . . . . . . . 11  |-  seq M
(  x.  ,  F
)  e.  _V
9 vex 2689 . . . . . . . . . . 11  |-  y  e. 
_V
108, 9breldm 4743 . . . . . . . . . 10  |-  (  seq M (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
117, 10syl6bi 162 . . . . . . . . 9  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
1211adantld 276 . . . . . . . 8  |-  ( n  =  M  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
13 eluzel2 9343 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1413, 3eleq2s 2234 . . . . . . . . . . . . . . . 16  |-  ( n  e.  Z  ->  M  e.  ZZ )
1514ad3antlr 484 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  - 
1 )  e.  Z
)  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  k  e.  Z )  ->  ( F `  k
)  e.  CC )
183, 15, 17prodf 11319 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F ) : Z --> CC )
19 simplr 519 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (
n  -  1 )  e.  Z )
2018, 19ffvelrnd 5556 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  (  seq M (  x.  ,  F ) `  (
n  -  1 ) )  e.  CC )
21 climcl 11063 . . . . . . . . . . . . . 14  |-  (  seq n (  x.  ,  F )  ~~>  y  -> 
y  e.  CC )
2221adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  y  e.  CC )
2320, 22mulcld 7798 . . . . . . . . . . . 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 9357 . . . . . . . . . . . . . . . . . . . 20  |-  ( ZZ>= `  M )  C_  ZZ
253, 24eqsstri 3129 . . . . . . . . . . . . . . . . . . 19  |-  Z  C_  ZZ
26 simplr 519 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  Z )
2725, 26sseldi 3095 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  ZZ )
2827zcnd 9186 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  n  e.  CC )
29 1cnd 7794 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
1  e.  CC )
3028, 29npcand 8089 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
( ( n  - 
1 )  +  1 )  =  n )
3130seqeq1d 10236 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  ->  seq ( ( n  - 
1 )  +  1 ) (  x.  ,  F )  =  seq n (  x.  ,  F ) )
3231breq1d 3939 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  Z )  /\  (
n  -  1 )  e.  Z )  -> 
(  seq ( ( n  -  1 )  +  1 ) (  x.  ,  F )  ~~>  y  <->  seq n
(  x.  ,  F
)  ~~>  y ) )
3332biimpar 295 . . . . . . . . . . . . 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 11320 . . . . . . . . . . . 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 4745 . . . . . . . . . . . 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 1320 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  ( n  -  1 )  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3736an32s 557 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( n  - 
1 )  e.  Z
)  ->  seq M (  x.  ,  F )  e.  dom  ~~>  )
3837expcom 115 . . . . . . . . 9  |-  ( ( n  -  1 )  e.  Z  ->  (
( ( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
393eqcomi 2143 . . . . . . . . 9  |-  ( ZZ>= `  M )  =  Z
4038, 39eleq2s 2234 . . . . . . . 8  |-  ( ( n  -  1 )  e.  ( ZZ>= `  M
)  ->  ( (
( ph  /\  n  e.  Z )  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
4112, 40jaoi 705 . . . . . . 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 114 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq n (  x.  ,  F )  ~~>  y  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4443adantld 276 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  (
( y #  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e. 
dom 
~~>  ) )
4544exlimdv 1791 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y )  ->  seq M (  x.  ,  F )  e.  dom  ~~>  ) )
4645rexlimdva 2549 . 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 103    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   _Vcvv 2686   class class class wbr 3929   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637    - cmin 7945   # cap 8355   ZZcz 9066   ZZ>=cuz 9338    seqcseq 10230    ~~> cli 11059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060
This theorem is referenced by: (None)
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