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Theorem ntrivcvgap0 11325
Description: A product that converges to a value apart from zero 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 )
ntrivcvgap0.4  |-  ( ph  ->  X #  0 )
Assertion
Ref Expression
ntrivcvgap0  |-  ( 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 ntrivcvgap0
StepHypRef Expression
1 ntrivcvgn0.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
2 uzid 9347 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 14 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 ntrivcvgn0.1 . . 3  |-  Z  =  ( ZZ>= `  M )
53, 4eleqtrrdi 2233 . 2  |-  ( ph  ->  M  e.  Z )
6 ntrivcvgn0.3 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  X )
7 climrel 11056 . . . . 5  |-  Rel  ~~>
87brrelex2i 4583 . . . 4  |-  (  seq M (  x.  ,  F )  ~~>  X  ->  X  e.  _V )
96, 8syl 14 . . 3  |-  ( ph  ->  X  e.  _V )
10 ntrivcvgap0.4 . . . 4  |-  ( ph  ->  X #  0 )
1110, 6jca 304 . . 3  |-  ( ph  ->  ( X #  0  /\ 
seq M (  x.  ,  F )  ~~>  X ) )
12 breq1 3932 . . . 4  |-  ( y  =  X  ->  (
y #  0  <->  X #  0
) )
13 breq2 3933 . . . 4  |-  ( y  =  X  ->  (  seq M (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  X ) )
1412, 13anbi12d 464 . . 3  |-  ( y  =  X  ->  (
( y #  0  /\ 
seq M (  x.  ,  F )  ~~>  y )  <-> 
( X #  0  /\ 
seq M (  x.  ,  F )  ~~>  X ) ) )
159, 11, 14elabd 2829 . 2  |-  ( ph  ->  E. y ( y #  0  /\  seq M
(  x.  ,  F
)  ~~>  y ) )
16 seqeq1 10228 . . . . . 6  |-  ( n  =  M  ->  seq n (  x.  ,  F )  =  seq M (  x.  ,  F ) )
1716breq1d 3939 . . . . 5  |-  ( n  =  M  ->  (  seq n (  x.  ,  F )  ~~>  y  <->  seq M (  x.  ,  F )  ~~>  y ) )
1817anbi2d 459 . . . 4  |-  ( n  =  M  ->  (
( y #  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  <-> 
( y #  0  /\ 
seq M (  x.  ,  F )  ~~>  y ) ) )
1918exbidv 1797 . . 3  |-  ( n  =  M  ->  ( E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y )  <->  E. y
( y #  0  /\ 
seq M (  x.  ,  F )  ~~>  y ) ) )
2019rspcev 2789 . 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 ) )
215, 15, 20syl2anc 408 1  |-  ( ph  ->  E. n  e.  Z  E. y ( y #  0  /\  seq n (  x.  ,  F )  ~~>  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   _Vcvv 2686   class class class wbr 3929   ` cfv 5123   0cc0 7627    x. cmul 7632   # cap 8350   ZZcz 9061   ZZ>=cuz 9333    seqcseq 10225    ~~> cli 11054
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-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-pre-ltirr 7739
This theorem depends on definitions:  df-bi 116  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-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-neg 7943  df-z 9062  df-uz 9334  df-seqfrec 10226  df-clim 11055
This theorem is referenced by: (None)
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