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Theorem clim1fr1 27636
Description: A class of sequences of fractions that converge to 1 (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
clim1fr1.1  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
clim1fr1.2  |-  ( ph  ->  A  e.  CC )
clim1fr1.3  |-  ( ph  ->  A  =/=  0 )
clim1fr1.4  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
clim1fr1  |-  ( ph  ->  F  ~~>  1 )
Distinct variable groups:    ph, n    A, n    B, n
Allowed substitution hint:    F( n)

Proof of Theorem clim1fr1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nnuz 10505 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10295 . . . 4  |-  1  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
4 nnex 9990 . . . . . 6  |-  NN  e.  _V
54mptex 5952 . . . . 5  |-  ( n  e.  NN  |->  1 )  e.  _V
65a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 )  e.  _V )
73zcnd 10360 . . . 4  |-  ( ph  ->  1  e.  CC )
8 eqidd 2431 . . . . . 6  |-  ( k  e.  NN  ->  (
n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 ) )
9 eqidd 2431 . . . . . 6  |-  ( ( k  e.  NN  /\  n  =  k )  ->  1  =  1 )
10 id 20 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN )
11 ax-1cn 9032 . . . . . . 7  |-  1  e.  CC
1211a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  1  e.  CC )
138, 9, 10, 12fvmptd 5796 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  1 ) `  k
)  =  1 )
1413adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  =  1 )
151, 3, 6, 7, 14climconst 12320 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  1 )  ~~>  1 )
16 clim1fr1.1 . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
174mptex 5952 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( A  x.  n
)  +  B )  /  ( A  x.  n ) ) )  e.  _V
1816, 17eqeltri 2500 . . . 4  |-  F  e. 
_V
1918a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
20 clim1fr1.4 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2120adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  B  e.  CC )
22 clim1fr1.2 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
24 nncn 9992 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  CC )
2524adantl 453 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
26 clim1fr1.3 . . . . . . 7  |-  ( ph  ->  A  =/=  0 )
2726adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  =/=  0 )
28 nnne0 10016 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
2928adantl 453 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3021, 23, 25, 27, 29divdiv1d 9805 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( B  /  A )  /  n )  =  ( B  /  ( A  x.  n )
) )
3130mpteq2dva 4282 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  =  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) )
3220, 22, 26divcld 9774 . . . . 5  |-  ( ph  ->  ( B  /  A
)  e.  CC )
33 divcnv 12616 . . . . 5  |-  ( ( B  /  A )  e.  CC  ->  (
n  e.  NN  |->  ( ( B  /  A
)  /  n ) )  ~~>  0 )
3432, 33syl 16 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  ~~>  0 )
3531, 34eqbrtrrd 4221 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) )  ~~>  0 )
36 eqid 2430 . . . . . 6  |-  ( n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 )
3711a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
3836, 37fmpti 5878 . . . . 5  |-  ( n  e.  NN  |->  1 ) : NN --> CC
3938a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 ) : NN --> CC )
4039ffvelrnda 5856 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  e.  CC )
4123, 25mulcld 9092 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  e.  CC )
4223, 25, 27, 29mulne0d 9658 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  =/=  0 )
4321, 41, 42divcld 9774 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( A  x.  n ) )  e.  CC )
44 eqid 2430 . . . . 5  |-  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) )
4543, 44fmptd 5879 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) ) : NN --> CC )
4645ffvelrnda 5856 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  e.  CC )
4716a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) ) )
48 oveq2 6075 . . . . . . . 8  |-  ( n  =  k  ->  ( A  x.  n )  =  ( A  x.  k ) )
4948oveq1d 6082 . . . . . . 7  |-  ( n  =  k  ->  (
( A  x.  n
)  +  B )  =  ( ( A  x.  k )  +  B ) )
5049, 48oveq12d 6085 . . . . . 6  |-  ( n  =  k  ->  (
( ( A  x.  n )  +  B
)  /  ( A  x.  n ) )  =  ( ( ( A  x.  k )  +  B )  / 
( A  x.  k
) ) )
5150adantl 453 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
)  =  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) ) )
52 simpr 448 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5322adantr 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
5452nncnd 10000 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  CC )
5553, 54mulcld 9092 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  e.  CC )
5620adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  CC )
5755, 56addcld 9091 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  +  B )  e.  CC )
5826adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  A  =/=  0 )
5952nnne0d 10028 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  =/=  0 )
6053, 54, 58, 59mulne0d 9658 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  =/=  0 )
6157, 55, 60divcld 9774 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  e.  CC )
6247, 51, 52, 61fvmptd 5796 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( A  x.  k )  +  B )  /  ( A  x.  k )
) )
6355, 56, 55, 60divdird 9812 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( ( ( A  x.  k )  / 
( A  x.  k
) )  +  ( B  /  ( A  x.  k ) ) ) )
6455, 60dividd 9772 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  /  ( A  x.  k ) )  =  1 )
6564oveq1d 6082 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  /  ( A  x.  k ) )  +  ( B  / 
( A  x.  k
) ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6663, 65eqtrd 2462 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6714eqcomd 2435 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  1  =  ( ( n  e.  NN  |->  1 ) `  k ) )
68 eqidd 2431 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) )
69 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  ->  n  =  k )
7069oveq2d 6083 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( A  x.  n
)  =  ( A  x.  k ) )
7170oveq2d 6083 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( B  /  ( A  x.  n )
)  =  ( B  /  ( A  x.  k ) ) )
7256, 55, 60divcld 9774 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  e.  CC )
7368, 71, 52, 72fvmptd 5796 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  =  ( B  / 
( A  x.  k
) ) )
7473eqcomd 2435 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  =  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) )
7567, 74oveq12d 6085 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  +  ( B  / 
( A  x.  k
) ) )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
7662, 66, 753eqtrd 2466 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
771, 3, 15, 19, 35, 40, 46, 76climadd 12408 . 2  |-  ( ph  ->  F  ~~>  ( 1  +  0 ) )
7811addid1i 9237 . 2  |-  ( 1  +  0 )  =  1
7977, 78syl6breq 4238 1  |-  ( ph  ->  F  ~~>  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2593   _Vcvv 2943   class class class wbr 4199    e. cmpt 4253   -->wf 5436   ` cfv 5440  (class class class)co 6067   CCcc 8972   0cc0 8974   1c1 8975    + caddc 8977    x. cmul 8979    / cdiv 9661   NNcn 9984   ZZcz 10266    ~~> cli 12261
This theorem is referenced by:  wallispilem5  27727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-pm 7007  df-en 7096  df-dom 7097  df-sdom 7098  df-sup 7432  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-fl 11185  df-seq 11307  df-exp 11366  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-rlim 12266
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