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Theorem clim1fr1 29942
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 11010 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10790 . . . 4  |-  1  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
4 nnex 10442 . . . . . 6  |-  NN  e.  _V
54mptex 6060 . . . . 5  |-  ( n  e.  NN  |->  1 )  e.  _V
65a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 )  e.  _V )
73zcnd 10862 . . . 4  |-  ( ph  ->  1  e.  CC )
8 eqidd 2455 . . . . . 6  |-  ( k  e.  NN  ->  (
n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 ) )
9 eqidd 2455 . . . . . 6  |-  ( ( k  e.  NN  /\  n  =  k )  ->  1  =  1 )
10 id 22 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN )
11 ax-1cn 9454 . . . . . . 7  |-  1  e.  CC
1211a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  1  e.  CC )
138, 9, 10, 12fvmptd 5891 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  1 ) `  k
)  =  1 )
1413adantl 466 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  =  1 )
151, 3, 6, 7, 14climconst 13142 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  1 )  ~~>  1 )
16 clim1fr1.1 . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
174mptex 6060 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( A  x.  n
)  +  B )  /  ( A  x.  n ) ) )  e.  _V
1816, 17eqeltri 2538 . . . 4  |-  F  e. 
_V
1918a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
20 clim1fr1.4 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2120adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  B  e.  CC )
22 clim1fr1.2 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
24 nncn 10444 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  CC )
2524adantl 466 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
26 clim1fr1.3 . . . . . . 7  |-  ( ph  ->  A  =/=  0 )
2726adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  =/=  0 )
28 nnne0 10468 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3021, 23, 25, 27, 29divdiv1d 10252 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( B  /  A )  /  n )  =  ( B  /  ( A  x.  n )
) )
3130mpteq2dva 4489 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  =  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) )
3220, 22, 26divcld 10221 . . . . 5  |-  ( ph  ->  ( B  /  A
)  e.  CC )
33 divcnv 13437 . . . . 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 4425 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) )  ~~>  0 )
36 eqid 2454 . . . . . 6  |-  ( n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 )
3711a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
3836, 37fmpti 5978 . . . . 5  |-  ( n  e.  NN  |->  1 ) : NN --> CC
3938a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 ) : NN --> CC )
4039ffvelrnda 5955 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  e.  CC )
4123, 25mulcld 9520 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  e.  CC )
4223, 25, 27, 29mulne0d 10102 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  =/=  0 )
4321, 41, 42divcld 10221 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( A  x.  n ) )  e.  CC )
44 eqid 2454 . . . . 5  |-  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) )
4543, 44fmptd 5979 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) ) : NN --> CC )
4645ffvelrnda 5955 . . 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 6211 . . . . . . . 8  |-  ( n  =  k  ->  ( A  x.  n )  =  ( A  x.  k ) )
4948oveq1d 6218 . . . . . . 7  |-  ( n  =  k  ->  (
( A  x.  n
)  +  B )  =  ( ( A  x.  k )  +  B ) )
5049, 48oveq12d 6221 . . . . . 6  |-  ( n  =  k  ->  (
( ( A  x.  n )  +  B
)  /  ( A  x.  n ) )  =  ( ( ( A  x.  k )  +  B )  / 
( A  x.  k
) ) )
5150adantl 466 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
)  =  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) ) )
52 simpr 461 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5322adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
5452nncnd 10452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  CC )
5553, 54mulcld 9520 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  e.  CC )
5620adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  CC )
5755, 56addcld 9519 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  +  B )  e.  CC )
5826adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  A  =/=  0 )
5952nnne0d 10480 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  =/=  0 )
6053, 54, 58, 59mulne0d 10102 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  =/=  0 )
6157, 55, 60divcld 10221 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  e.  CC )
6247, 51, 52, 61fvmptd 5891 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( A  x.  k )  +  B )  /  ( A  x.  k )
) )
6355, 56, 55, 60divdird 10259 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( ( ( A  x.  k )  / 
( A  x.  k
) )  +  ( B  /  ( A  x.  k ) ) ) )
6455, 60dividd 10219 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  /  ( A  x.  k ) )  =  1 )
6564oveq1d 6218 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  /  ( A  x.  k ) )  +  ( B  / 
( A  x.  k
) ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6663, 65eqtrd 2495 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6714eqcomd 2462 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  1  =  ( ( n  e.  NN  |->  1 ) `  k ) )
68 eqidd 2455 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) )
69 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  ->  n  =  k )
7069oveq2d 6219 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( A  x.  n
)  =  ( A  x.  k ) )
7170oveq2d 6219 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( B  /  ( A  x.  n )
)  =  ( B  /  ( A  x.  k ) ) )
7256, 55, 60divcld 10221 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  e.  CC )
7368, 71, 52, 72fvmptd 5891 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  =  ( B  / 
( A  x.  k
) ) )
7473eqcomd 2462 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  =  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) )
7567, 74oveq12d 6221 . . . 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 2499 . . 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 13230 . 2  |-  ( ph  ->  F  ~~>  ( 1  +  0 ) )
78 1p0e1 10548 . 2  |-  ( 1  +  0 )  =  1
7977, 78syl6breq 4442 1  |-  ( ph  ->  F  ~~>  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    / cdiv 10107   NNcn 10436   ZZcz 10760    ~~> cli 13083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fl 11762  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-rlim 13088
This theorem is referenced by:  wallispilem5  30032
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