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Theorem ntrivcvgn0 27552
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 10981 . . . 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 2551 . 2  |-  ( ph  ->  M  e.  Z )
6 ntrivcvgn0.3 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
7 climrel 13083 . . . . 5  |-  Rel  ~~>
87brrelex2i 4983 . . . 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 532 . . 3  |-  ( ph  ->  ( X  =/=  0  /\  seq M (  x.  ,  F )  ~~>  X ) )
12 neeq1 2730 . . . . 5  |-  ( y  =  X  ->  (
y  =/=  0  <->  X  =/=  0 ) )
13 breq2 4399 . . . . 5  |-  ( y  =  X  ->  (  seq M (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  X ) )
1412, 13anbi12d 710 . . . 4  |-  ( y  =  X  ->  (
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y )  <-> 
( X  =/=  0  /\  seq M (  x.  ,  F )  ~~>  X ) ) )
1514spcegv 3158 . . 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 11921 . . . . . 6  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
1817breq1d 4405 . . . . 5  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
1918anbi2d 703 . . . 4  |-  ( n  =  M  ->  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  <-> 
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) ) )
2019exbidv 1681 . . 3  |-  ( n  =  M  ->  ( E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  <->  E. y
( y  =/=  0  /\  seq M (  x.  ,  F )  ~~>  y ) ) )
2120rspcev 3173 . 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 661 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 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2645   E.wrex 2797   _Vcvv 3072   class class class wbr 4395   ` cfv 5521   0cc0 9388    x. cmul 9393   ZZcz 10752   ZZ>=cuz 10967    seqcseq 11918    ~~> cli 13075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-pre-lttri 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-neg 9704  df-z 10753  df-uz 10968  df-seq 11919  df-clim 13079
This theorem is referenced by:  zprodn0  27591
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