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Theorem cvg1nlemf 10479
Description: Lemma for cvg1n 10482. The modified sequence  G is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
Hypotheses
Ref Expression
cvg1n.f  |-  ( ph  ->  F : NN --> RR )
cvg1n.c  |-  ( ph  ->  C  e.  RR+ )
cvg1n.cau  |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( F `  n )  <  ( ( F `
 k )  +  ( C  /  n
) )  /\  ( F `  k )  <  ( ( F `  n )  +  ( C  /  n ) ) ) )
cvg1nlem.g  |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )
cvg1nlem.z  |-  ( ph  ->  Z  e.  NN )
cvg1nlem.start  |-  ( ph  ->  C  <  Z )
Assertion
Ref Expression
cvg1nlemf  |-  ( ph  ->  G : NN --> RR )
Distinct variable group:    ph, j
Allowed substitution hints:    ph( k, n)    C( j, k, n)    F( j,
k, n)    G( j,
k, n)    Z( j,
k, n)

Proof of Theorem cvg1nlemf
StepHypRef Expression
1 cvg1n.f . . . 4  |-  ( ph  ->  F : NN --> RR )
21adantr 271 . . 3  |-  ( (
ph  /\  j  e.  NN )  ->  F : NN
--> RR )
3 simpr 109 . . . 4  |-  ( (
ph  /\  j  e.  NN )  ->  j  e.  NN )
4 cvg1nlem.z . . . . 5  |-  ( ph  ->  Z  e.  NN )
54adantr 271 . . . 4  |-  ( (
ph  /\  j  e.  NN )  ->  Z  e.  NN )
63, 5nnmulcld 8534 . . 3  |-  ( (
ph  /\  j  e.  NN )  ->  ( j  x.  Z )  e.  NN )
72, 6ffvelrnd 5451 . 2  |-  ( (
ph  /\  j  e.  NN )  ->  ( F `
 ( j  x.  Z ) )  e.  RR )
8 cvg1nlem.g . 2  |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )
97, 8fmptd 5468 1  |-  ( ph  ->  G : NN --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   A.wral 2360   class class class wbr 3853    |-> cmpt 3907   -->wf 5026   ` cfv 5030  (class class class)co 5668   RRcr 7412    + caddc 7416    x. cmul 7418    < clt 7585    / cdiv 8202   NNcn 8485   ZZ>=cuz 9082   RR+crp 9197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-1rid 7515  ax-cnre 7519
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fv 5038  df-ov 5671  df-inn 8486
This theorem is referenced by:  cvg1nlemres  10481
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