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Theorem faclimlem7 24123
Description: Lemma for faclim 24126. Sequence closure. (Contributed by Scott Fenton, 26-Nov-2017.)
Assertion
Ref Expression
faclimlem7  |-  ( ( B  e.  NN0  /\  K  e.  NN )  ->  (  seq  1 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  /  ( 1  +  ( B  /  n
) ) ) ) ) `  K )  e.  CC )
Distinct variable group:    B, n
Allowed substitution hint:    K( n)

Proof of Theorem faclimlem7
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnnuz 10280 . . . 4  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
21biimpi 186 . . 3  |-  ( K  e.  NN  ->  K  e.  ( ZZ>= `  1 )
)
32adantl 452 . 2  |-  ( ( B  e.  NN0  /\  K  e.  NN )  ->  K  e.  ( ZZ>= ` 
1 ) )
4 elfznn 10835 . . . 4  |-  ( m  e.  ( 1 ... K )  ->  m  e.  NN )
5 faclimlem2 24118 . . . . . . 7  |-  ( n  =  m  ->  (
( ( 1  +  ( 1  /  n
) ) ^ B
)  /  ( 1  +  ( B  /  n ) ) )  =  ( ( ( 1  +  ( 1  /  m ) ) ^ B )  / 
( 1  +  ( B  /  m ) ) ) )
6 eqid 2296 . . . . . . 7  |-  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  /  ( 1  +  ( B  /  n
) ) ) )  =  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  / 
( 1  +  ( B  /  n ) ) ) )
7 ovex 5899 . . . . . . 7  |-  ( ( ( 1  +  ( 1  /  m ) ) ^ B )  /  ( 1  +  ( B  /  m
) ) )  e. 
_V
85, 6, 7fvmpt 5618 . . . . . 6  |-  ( m  e.  NN  ->  (
( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  /  (
1  +  ( B  /  n ) ) ) ) `  m
)  =  ( ( ( 1  +  ( 1  /  m ) ) ^ B )  /  ( 1  +  ( B  /  m
) ) ) )
98adantl 452 . . . . 5  |-  ( ( B  e.  NN0  /\  m  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  / 
( 1  +  ( B  /  n ) ) ) ) `  m )  =  ( ( ( 1  +  ( 1  /  m
) ) ^ B
)  /  ( 1  +  ( B  /  m ) ) ) )
10 faclimlem6 24122 . . . . 5  |-  ( ( B  e.  NN0  /\  m  e.  NN )  ->  ( ( ( 1  +  ( 1  /  m ) ) ^ B )  /  (
1  +  ( B  /  m ) ) )  e.  CC )
119, 10eqeltrd 2370 . . . 4  |-  ( ( B  e.  NN0  /\  m  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  / 
( 1  +  ( B  /  n ) ) ) ) `  m )  e.  CC )
124, 11sylan2 460 . . 3  |-  ( ( B  e.  NN0  /\  m  e.  ( 1 ... K ) )  ->  ( ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  /  ( 1  +  ( B  /  n
) ) ) ) `
 m )  e.  CC )
1312adantlr 695 . 2  |-  ( ( ( B  e.  NN0  /\  K  e.  NN )  /\  m  e.  ( 1 ... K ) )  ->  ( (
n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n
) ) ^ B
)  /  ( 1  +  ( B  /  n ) ) ) ) `  m )  e.  CC )
14 mulcl 8837 . . 3  |-  ( ( m  e.  CC  /\  k  e.  CC )  ->  ( m  x.  k
)  e.  CC )
1514adantl 452 . 2  |-  ( ( ( B  e.  NN0  /\  K  e.  NN )  /\  ( m  e.  CC  /\  k  e.  CC ) )  -> 
( m  x.  k
)  e.  CC )
163, 13, 15seqcl 11082 1  |-  ( ( B  e.  NN0  /\  K  e.  NN )  ->  (  seq  1 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ B )  /  ( 1  +  ( B  /  n
) ) ) ) ) `  K )  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   ^cexp 11120
This theorem is referenced by:  faclimlem9  24125  faclim  24126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121
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