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Theorem trilpolemclim 14720
Description: Lemma for trilpo 14727. 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 5882 . . . . . 6  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
32oveq2d 5890 . . . . 5  |-  ( n  =  k  ->  (
1  /  ( 2 ^ n ) )  =  ( 1  / 
( 2 ^ k
) ) )
4 fveq2 5515 . . . . 5  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
53, 4oveq12d 5892 . . . 4  |-  ( n  =  k  ->  (
( 1  /  (
2 ^ n ) )  x.  ( F `
 n ) )  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k
) ) )
6 simpr 110 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
7 2rp 9657 . . . . . . . . 9  |-  2  e.  RR+
87a1i 9 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  2  e.  RR+ )
96nnzd 9373 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ZZ )
108, 9rpexpcld 10677 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2 ^ k )  e.  RR+ )
1110rpreccld 9706 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  RR+ )
1211rpred 9695 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  RR )
13 simpr 110 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( F `  k
)  =  0 )
14 0re 7956 . . . . . . 7  |-  0  e.  RR
1513, 14eqeltrdi 2268 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( F `  k
)  e.  RR )
16 simpr 110 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( F `  k
)  =  1 )
17 1re 7955 . . . . . . 7  |-  1  e.  RR
1816, 17eqeltrdi 2268 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( F `  k
)  e.  RR )
19 trilpolemgt1.f . . . . . . . 8  |-  ( ph  ->  F : NN --> { 0 ,  1 } )
2019ffvelcdmda 5651 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
{ 0 ,  1 } )
21 elpri 3615 . . . . . . 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 798 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  RR )
2412, 23remulcld 7987 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) )  e.  RR )
251, 5, 6, 24fvmptd3 5609 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  =  ( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
) )
2625, 24eqeltrd 2254 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
2711rpge0d 9699 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( 1  /  (
2 ^ k ) ) )
28 0le0 9007 . . . . . 6  |-  0  <_  0
2928, 13breqtrrid 4041 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
0  <_  ( F `  k ) )
30 0le1 8437 . . . . . 6  |-  0  <_  1
3130, 16breqtrrid 4041 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
0  <_  ( F `  k ) )
3229, 31, 22mpjaodan 798 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( F `  k
) )
3312, 23, 27, 32mulge0d 8577 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
) )
3433, 25breqtrrd 4031 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( G `  k
) )
3525adantr 276 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) ) )
3613oveq2d 5890 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  0 ) )
3711rpcnd 9697 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  /  ( 2 ^ k ) )  e.  CC )
3837adantr 276 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( 1  /  (
2 ^ k ) )  e.  CC )
3938mul01d 8349 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  0 )  =  0 )
4035, 36, 393eqtrd 2214 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  =  0 )
4127adantr 276 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
0  <_  ( 1  /  ( 2 ^ k ) ) )
4240, 41eqbrtrd 4025 . . 3  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  0 )  -> 
( G `  k
)  <_  ( 1  /  ( 2 ^ k ) ) )
4325adantr 276 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  ( F `  k ) ) )
4416oveq2d 5890 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  ( F `  k )
)  =  ( ( 1  /  ( 2 ^ k ) )  x.  1 ) )
4537adantr 276 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  e.  CC )
4645mulridd 7973 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( ( 1  / 
( 2 ^ k
) )  x.  1 )  =  ( 1  /  ( 2 ^ k ) ) )
4743, 44, 463eqtrd 2214 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  =  ( 1  /  ( 2 ^ k ) ) )
4812adantr 276 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  e.  RR )
4948leidd 8470 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( 1  /  (
2 ^ k ) )  <_  ( 1  /  ( 2 ^ k ) ) )
5047, 49eqbrtrd 4025 . . 3  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( F `  k )  =  1 )  -> 
( G `  k
)  <_  ( 1  /  ( 2 ^ k ) ) )
5142, 50, 22mpjaodan 798 . 2  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  <_ 
( 1  /  (
2 ^ k ) ) )
5226, 34, 51cvgcmp2n 14717 1  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148   {cpr 3593   class class class wbr 4003    |-> cmpt 4064   dom cdm 4626   -->wf 5212   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811    + caddc 7813    x. cmul 7815    <_ cle 7992    / cdiv 8628   NNcn 8918   2c2 8969   RR+crp 9652    seqcseq 10444   ^cexp 10518    ~~> cli 11285
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-ico 9893  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-sumdc 11361
This theorem is referenced by:  trilpolemcl  14721  trilpolemisumle  14722  trilpolemeq1  14724  trilpolemlt1  14725  nconstwlpolemgt0  14747
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