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Theorem basellem7 20547
Description: Lemma for basel 20550. The function  1  +  A  x.  G for any fixed  A goes to  1. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
basel.g  |-  G  =  ( n  e.  NN  |->  ( 1  /  (
( 2  x.  n
)  +  1 ) ) )
basellem7.2  |-  A  e.  CC
Assertion
Ref Expression
basellem7  |-  ( ( NN  X.  { 1 } )  o F  +  ( ( NN 
X.  { A }
)  o F  x.  G ) )  ~~>  1

Proof of Theorem basellem7
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 10414 . . . 4  |-  NN  =  ( ZZ>= `  1 )
2 1z 10204 . . . . 5  |-  1  e.  ZZ
32a1i 10 . . . 4  |-  (  T. 
->  1  e.  ZZ )
4 ax-1cn 8942 . . . . 5  |-  1  e.  CC
51eqimss2i 3319 . . . . . 6  |-  ( ZZ>= ` 
1 )  C_  NN
6 nnex 9899 . . . . . 6  |-  NN  e.  _V
75, 6climconst2 12229 . . . . 5  |-  ( ( 1  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  {
1 } )  ~~>  1 )
84, 3, 7sylancr 644 . . . 4  |-  (  T. 
->  ( NN  X.  {
1 } )  ~~>  1 )
9 ovex 6006 . . . . 5  |-  ( ( NN  X.  { 1 } )  o F  +  ( ( NN 
X.  { A }
)  o F  x.  G ) )  e. 
_V
109a1i 10 . . . 4  |-  (  T. 
->  ( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { A } )  o F  x.  G ) )  e.  _V )
11 basellem7.2 . . . . . . 7  |-  A  e.  CC
125, 6climconst2 12229 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { A } )  ~~>  A )
1311, 3, 12sylancr 644 . . . . . 6  |-  (  T. 
->  ( NN  X.  { A } )  ~~>  A )
14 ovex 6006 . . . . . . 7  |-  ( ( NN  X.  { A } )  o F  x.  G )  e. 
_V
1514a1i 10 . . . . . 6  |-  (  T. 
->  ( ( NN  X.  { A } )  o F  x.  G )  e.  _V )
16 basel.g . . . . . . . 8  |-  G  =  ( n  e.  NN  |->  ( 1  /  (
( 2  x.  n
)  +  1 ) ) )
1716basellem6 20546 . . . . . . 7  |-  G  ~~>  0
1817a1i 10 . . . . . 6  |-  (  T. 
->  G  ~~>  0 )
1911elexi 2882 . . . . . . . . 9  |-  A  e. 
_V
2019fconst 5533 . . . . . . . 8  |-  ( NN 
X.  { A }
) : NN --> { A }
2111a1i 10 . . . . . . . . 9  |-  (  T. 
->  A  e.  CC )
2221snssd 3858 . . . . . . . 8  |-  (  T. 
->  { A }  C_  CC )
23 fss 5503 . . . . . . . 8  |-  ( ( ( NN  X.  { A } ) : NN --> { A }  /\  { A }  C_  CC )  ->  ( NN  X.  { A } ) : NN --> CC )
2420, 22, 23sylancr 644 . . . . . . 7  |-  (  T. 
->  ( NN  X.  { A } ) : NN --> CC )
25 ffvelrn 5770 . . . . . . 7  |-  ( ( ( NN  X.  { A } ) : NN --> CC  /\  k  e.  NN )  ->  ( ( NN 
X.  { A }
) `  k )  e.  CC )
2624, 25sylan 457 . . . . . 6  |-  ( (  T.  /\  k  e.  NN )  ->  (
( NN  X.  { A } ) `  k
)  e.  CC )
27 2nn 10026 . . . . . . . . . . . . 13  |-  2  e.  NN
2827a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  2  e.  NN )
29 nnmulcl 9916 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN  /\  n  e.  NN )  ->  ( 2  x.  n
)  e.  NN )
3028, 29sylan 457 . . . . . . . . . . 11  |-  ( (  T.  /\  n  e.  NN )  ->  (
2  x.  n )  e.  NN )
3130peano2nnd 9910 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  NN )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
3231nnrecred 9938 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  NN )  ->  (
1  /  ( ( 2  x.  n )  +  1 ) )  e.  RR )
3332recnd 9008 . . . . . . . 8  |-  ( (  T.  /\  n  e.  NN )  ->  (
1  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3433, 16fmptd 5795 . . . . . . 7  |-  (  T. 
->  G : NN --> CC )
35 ffvelrn 5770 . . . . . . 7  |-  ( ( G : NN --> CC  /\  k  e.  NN )  ->  ( G `  k
)  e.  CC )
3634, 35sylan 457 . . . . . 6  |-  ( (  T.  /\  k  e.  NN )  ->  ( G `  k )  e.  CC )
37 ffn 5495 . . . . . . . 8  |-  ( ( NN  X.  { A } ) : NN --> CC  ->  ( NN  X.  { A } )  Fn  NN )
3824, 37syl 15 . . . . . . 7  |-  (  T. 
->  ( NN  X.  { A } )  Fn  NN )
39 ffn 5495 . . . . . . . 8  |-  ( G : NN --> CC  ->  G  Fn  NN )
4034, 39syl 15 . . . . . . 7  |-  (  T. 
->  G  Fn  NN )
416a1i 10 . . . . . . 7  |-  (  T. 
->  NN  e.  _V )
42 inidm 3466 . . . . . . 7  |-  ( NN 
i^i  NN )  =  NN
43 eqidd 2367 . . . . . . 7  |-  ( (  T.  /\  k  e.  NN )  ->  (
( NN  X.  { A } ) `  k
)  =  ( ( NN  X.  { A } ) `  k
) )
44 eqidd 2367 . . . . . . 7  |-  ( (  T.  /\  k  e.  NN )  ->  ( G `  k )  =  ( G `  k ) )
4538, 40, 41, 41, 42, 43, 44ofval 6214 . . . . . 6  |-  ( (  T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { A } )  o F  x.  G ) `
 k )  =  ( ( ( NN 
X.  { A }
) `  k )  x.  ( G `  k
) ) )
461, 3, 13, 15, 18, 26, 36, 45climmul 12313 . . . . 5  |-  (  T. 
->  ( ( NN  X.  { A } )  o F  x.  G )  ~~>  ( A  x.  0 ) )
4711mul01i 9149 . . . . 5  |-  ( A  x.  0 )  =  0
4846, 47syl6breq 4164 . . . 4  |-  (  T. 
->  ( ( NN  X.  { A } )  o F  x.  G )  ~~>  0 )
49 1ex 8980 . . . . . . 7  |-  1  e.  _V
5049fconst 5533 . . . . . 6  |-  ( NN 
X.  { 1 } ) : NN --> { 1 }
514a1i 10 . . . . . . 7  |-  (  T. 
->  1  e.  CC )
5251snssd 3858 . . . . . 6  |-  (  T. 
->  { 1 }  C_  CC )
53 fss 5503 . . . . . 6  |-  ( ( ( NN  X.  {
1 } ) : NN --> { 1 }  /\  { 1 } 
C_  CC )  -> 
( NN  X.  {
1 } ) : NN --> CC )
5450, 52, 53sylancr 644 . . . . 5  |-  (  T. 
->  ( NN  X.  {
1 } ) : NN --> CC )
55 ffvelrn 5770 . . . . 5  |-  ( ( ( NN  X.  {
1 } ) : NN --> CC  /\  k  e.  NN )  ->  (
( NN  X.  {
1 } ) `  k )  e.  CC )
5654, 55sylan 457 . . . 4  |-  ( (  T.  /\  k  e.  NN )  ->  (
( NN  X.  {
1 } ) `  k )  e.  CC )
57 mulcl 8968 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
5857adantl 452 . . . . . 6  |-  ( (  T.  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
5958, 24, 34, 41, 41, 42off 6220 . . . . 5  |-  (  T. 
->  ( ( NN  X.  { A } )  o F  x.  G ) : NN --> CC )
60 ffvelrn 5770 . . . . 5  |-  ( ( ( ( NN  X.  { A } )  o F  x.  G ) : NN --> CC  /\  k  e.  NN )  ->  ( ( ( NN 
X.  { A }
)  o F  x.  G ) `  k
)  e.  CC )
6159, 60sylan 457 . . . 4  |-  ( (  T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { A } )  o F  x.  G ) `
 k )  e.  CC )
6250a1i 10 . . . . . 6  |-  (  T. 
->  ( NN  X.  {
1 } ) : NN --> { 1 } )
63 ffn 5495 . . . . . 6  |-  ( ( NN  X.  { 1 } ) : NN --> { 1 }  ->  ( NN  X.  { 1 } )  Fn  NN )
6462, 63syl 15 . . . . 5  |-  (  T. 
->  ( NN  X.  {
1 } )  Fn  NN )
65 ffn 5495 . . . . . 6  |-  ( ( ( NN  X.  { A } )  o F  x.  G ) : NN --> CC  ->  (
( NN  X.  { A } )  o F  x.  G )  Fn  NN )
6659, 65syl 15 . . . . 5  |-  (  T. 
->  ( ( NN  X.  { A } )  o F  x.  G )  Fn  NN )
67 eqidd 2367 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  (
( NN  X.  {
1 } ) `  k )  =  ( ( NN  X.  {
1 } ) `  k ) )
68 eqidd 2367 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { A } )  o F  x.  G ) `
 k )  =  ( ( ( NN 
X.  { A }
)  o F  x.  G ) `  k
) )
6964, 66, 41, 41, 42, 67, 68ofval 6214 . . . 4  |-  ( (  T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { A } )  o F  x.  G ) ) `
 k )  =  ( ( ( NN 
X.  { 1 } ) `  k )  +  ( ( ( NN  X.  { A } )  o F  x.  G ) `  k ) ) )
701, 3, 8, 10, 48, 56, 61, 69climadd 12312 . . 3  |-  (  T. 
->  ( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { A } )  o F  x.  G ) )  ~~>  ( 1  +  0 ) )
7170trud 1328 . 2  |-  ( ( NN  X.  { 1 } )  o F  +  ( ( NN 
X.  { A }
)  o F  x.  G ) )  ~~>  ( 1  +  0 )
724addid1i 9146 . 2  |-  ( 1  +  0 )  =  1
7371, 72breqtri 4148 1  |-  ( ( NN  X.  { 1 } )  o F  +  ( ( NN 
X.  { A }
)  o F  x.  G ) )  ~~>  1
Colors of variables: wff set class
Syntax hints:    /\ wa 358    T. wtru 1321    = wceq 1647    e. wcel 1715   _Vcvv 2873    C_ wss 3238   {csn 3729   class class class wbr 4125    e. cmpt 4179    X. cxp 4790    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981    o Fcof 6203   CCcc 8882   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    / cdiv 9570   NNcn 9893   2c2 9942   ZZcz 10175   ZZ>=cuz 10381    ~~> cli 12165
This theorem is referenced by:  basellem9  20549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fl 11089  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170
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