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Theorem cvg1nlemf 11033
Description: Lemma for cvg1n 11036. 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 276 . . 3  |-  ( (
ph  /\  j  e.  NN )  ->  F : NN
--> RR )
3 simpr 110 . . . 4  |-  ( (
ph  /\  j  e.  NN )  ->  j  e.  NN )
4 cvg1nlem.z . . . . 5  |-  ( ph  ->  Z  e.  NN )
54adantr 276 . . . 4  |-  ( (
ph  /\  j  e.  NN )  ->  Z  e.  NN )
63, 5nnmulcld 9003 . . 3  |-  ( (
ph  /\  j  e.  NN )  ->  ( j  x.  Z )  e.  NN )
72, 6ffvelcdmd 5676 . 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 5694 1  |-  ( ph  ->  G : NN --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   class class class wbr 4021    |-> cmpt 4082   -->wf 5234   ` cfv 5238  (class class class)co 5900   RRcr 7845    + caddc 7849    x. cmul 7851    < clt 8027    / cdiv 8664   NNcn 8954   ZZ>=cuz 9563   RR+crp 9689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-1rid 7953  ax-cnre 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fv 5246  df-ov 5903  df-inn 8955
This theorem is referenced by:  cvg1nlemres  11035
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