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Theorem cvg1nlemf 10767
Description: Lemma for cvg1n 10770. 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 274 . . 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 274 . . . 4  |-  ( (
ph  /\  j  e.  NN )  ->  Z  e.  NN )
63, 5nnmulcld 8781 . . 3  |-  ( (
ph  /\  j  e.  NN )  ->  ( j  x.  Z )  e.  NN )
72, 6ffvelrnd 5556 . 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 5574 1  |-  ( ph  ->  G : NN --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   class class class wbr 3929    |-> cmpt 3989   -->wf 5119   ` cfv 5123  (class class class)co 5774   RRcr 7631    + caddc 7635    x. cmul 7637    < clt 7812    / cdiv 8444   NNcn 8732   ZZ>=cuz 9338   RR+crp 9453
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-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-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-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-1rid 7739  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  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-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  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-int 3772  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-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-inn 8733
This theorem is referenced by:  cvg1nlemres  10769
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