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Theorem trilpolemclim 13229
Description: Lemma for trilpo 13236. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
Hypotheses
Ref Expression
trilpolemgt1.f  |-  ( ph  ->  F : NN --> { 0 ,  1 } )
trilpolemclim.g  |-  G  =  ( n  e.  NN  |->  ( ( 1  / 
( 2 ^ n
) )  x.  ( F `  n )
) )
Assertion
Ref Expression
trilpolemclim  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
Distinct variable group:    n, F
Allowed substitution hints:    ph( n)    G( n)

Proof of Theorem trilpolemclim
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 trilpolemclim.g . . . 4  |-  G  =  ( n  e.  NN  |->  ( ( 1  / 
( 2 ^ n
) )  x.  ( F `  n )
) )
2 oveq2 5782 . . . . . 6  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
32oveq2d 5790 . . . . 5  |-  ( n  =  k  ->  (
1  /  ( 2 ^ n ) )  =  ( 1  / 
( 2 ^ k
) ) )
4 fveq2 5421 . . . . 5  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
53, 4oveq12d 5792 . . . 4  |-  ( n  =  k  ->  (
( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) )  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k
) ) )
6 simpr 109 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
7 2rp 9446 . . . . . . . . 9  |-  2  e.  RR+
87a1i 9 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  2  e.  RR+ )
96nnzd 9172 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ZZ )
108, 9rpexpcld 10448 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2 ^ k )  e.  RR+ )
1110rpreccld 9494 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  RR+ )
1211rpred 9483 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  RR )
13 simpr 109 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( F `  k
)  =  0 )
14 0re 7766 . . . . . . 7  |-  0  e.  RR
1513, 14eqeltrdi 2230 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( F `  k
)  e.  RR )
16 simpr 109 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( F `  k
)  =  1 )
17 1re 7765 . . . . . . 7  |-  1  e.  RR
1816, 17eqeltrdi 2230 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( F `  k
)  e.  RR )
19 trilpolemgt1.f . . . . . . . 8  |-  ( ph  ->  F : NN --> { 0 ,  1 } )
2019ffvelrnda 5555 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
{ 0 ,  1 } )
21 elpri 3550 . . . . . . 7  |-  ( ( F `  k )  e.  { 0 ,  1 }  ->  (
( F `  k
)  =  0  \/  ( F `  k
)  =  1 ) )
2220, 21syl 14 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  =  0  \/  ( F `  k )  =  1 ) )
2315, 18, 22mpjaodan 787 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  RR )
2412, 23remulcld 7796 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) )  e.  RR )
251, 5, 6, 24fvmptd3 5514 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  =  ( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
) )
2625, 24eqeltrd 2216 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
2711rpge0d 9487 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( 1  /  (
2 ^ k ) ) )
28 0le0 8809 . . . . . 6  |-  0  <_  0
2928, 13breqtrrid 3966 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
0  <_  ( F `  k ) )
30 0le1 8243 . . . . . 6  |-  0  <_  1
3130, 16breqtrrid 3966 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
0  <_  ( F `  k ) )
3229, 31, 22mpjaodan 787 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( F `  k
) )
3312, 23, 27, 32mulge0d 8383 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
) )
3433, 25breqtrrd 3956 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( G `  k
) )
3525adantr 274 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) ) )
3613oveq2d 5790 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  0 ) )
3711rpcnd 9485 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  CC )
3837adantr 274 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( 1  /  (
2 ^ k ) )  e.  CC )
3938mul01d 8155 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  0 )  =  0 )
4035, 36, 393eqtrd 2176 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  =  0 )
4127adantr 274 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
0  <_  ( 1  /  ( 2 ^ k ) ) )
4240, 41eqbrtrd 3950 . . 3  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  <_  ( 1  /  ( 2 ^ k ) ) )
4325adantr 274 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) ) )
4416oveq2d 5790 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  1 ) )
4537adantr 274 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  e.  CC )
4645mulid1d 7783 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  1 )  =  ( 1  /  ( 2 ^ k ) ) )
4743, 44, 463eqtrd 2176 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  =  ( 1  /  ( 2 ^ k ) ) )
4812adantr 274 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  e.  RR )
4948leidd 8276 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  <_  ( 1  /  ( 2 ^ k ) ) )
5047, 49eqbrtrd 3950 . . 3  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  <_  ( 1  /  ( 2 ^ k ) ) )
5142, 50, 22mpjaodan 787 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  <_ 
( 1  /  (
2 ^ k ) ) )
5226, 34, 51cvgcmp2n 13228 1  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   {cpr 3528   class class class wbr 3929    |-> cmpt 3989   dom cdm 4539   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621    + caddc 7623    x. cmul 7625    <_ cle 7801    / cdiv 8432   NNcn 8720   2c2 8771   RR+crp 9441    seqcseq 10218   ^cexp 10292    ~~> cli 11047
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-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740
This theorem depends on definitions:  df-bi 116  df-dc 820  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-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  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-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123
This theorem is referenced by:  trilpolemcl  13230  trilpolemisumle  13231  trilpolemeq1  13233  trilpolemlt1  13234
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