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Theorem ntrivcvgn0 13792
Description: A product that converges to a non-zero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgn0.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvgn0.2  |-  ( ph  ->  M  e.  ZZ )
ntrivcvgn0.3  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
ntrivcvgn0.4  |-  ( ph  ->  X  =/=  0 )
Assertion
Ref Expression
ntrivcvgn0  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
Distinct variable groups:    n, F, y    n, M, y    y, X    n, Z
Allowed substitution hints:    ph( y, n)    X( n)    Z( y)

Proof of Theorem ntrivcvgn0
StepHypRef Expression
1 ntrivcvgn0.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
2 uzid 11096 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 16 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 ntrivcvgn0.1 . . 3  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2553 . 2  |-  ( ph  ->  M  e.  Z )
6 ntrivcvgn0.3 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
7 climrel 13400 . . . . 5  |-  Rel  ~~>
87brrelex2i 5030 . . . 4  |-  (  seq M (  x.  ,  F )  ~~>  X  ->  X  e.  _V )
96, 8syl 16 . . 3  |-  ( ph  ->  X  e.  _V )
10 ntrivcvgn0.4 . . . 4  |-  ( ph  ->  X  =/=  0 )
1110, 6jca 530 . . 3  |-  ( ph  ->  ( X  =/=  0  /\  seq M (  x.  ,  F )  ~~>  X ) )
12 neeq1 2735 . . . . 5  |-  ( y  =  X  ->  (
y  =/=  0  <->  X  =/=  0 ) )
13 breq2 4443 . . . . 5  |-  ( y  =  X  ->  (  seq M (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  X ) )
1412, 13anbi12d 708 . . . 4  |-  ( y  =  X  ->  (
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y )  <-> 
( X  =/=  0  /\  seq M (  x.  ,  F )  ~~>  X ) ) )
1514spcegv 3192 . . 3  |-  ( X  e.  _V  ->  (
( X  =/=  0  /\  seq M (  x.  ,  F )  ~~>  X )  ->  E. y ( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) ) )
169, 11, 15sylc 60 . 2  |-  ( ph  ->  E. y ( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) )
17 seqeq1 12095 . . . . . 6  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
1817breq1d 4449 . . . . 5  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
1918anbi2d 701 . . . 4  |-  ( n  =  M  ->  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  <-> 
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) ) )
2019exbidv 1719 . . 3  |-  ( n  =  M  ->  ( E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  <->  E. y
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) ) )
2120rspcev 3207 . 2  |-  ( ( M  e.  Z  /\  E. y ( y  =/=  0  /\  seq M
(  x.  ,  F
)  ~~>  y ) )  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
225, 16, 21syl2anc 659 1  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   _Vcvv 3106   class class class wbr 4439   ` cfv 5570   0cc0 9481    x. cmul 9486   ZZcz 10860   ZZ>=cuz 11082    seqcseq 12092    ~~> cli 13392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9799  df-z 10861  df-uz 11083  df-seq 12093  df-clim 13396
This theorem is referenced by:  zprodn0  13831
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