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Theorem clim1fr1 27829
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 10279 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10069 . . . 4  |-  1  e.  ZZ
32a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
4 nnex 9768 . . . . . 6  |-  NN  e.  _V
54mptex 5762 . . . . 5  |-  ( n  e.  NN  |->  1 )  e.  _V
65a1i 10 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 )  e.  _V )
73zcnd 10134 . . . 4  |-  ( ph  ->  1  e.  CC )
8 eqidd 2297 . . . . . 6  |-  ( k  e.  NN  ->  (
n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 ) )
9 eqidd 2297 . . . . . 6  |-  ( ( k  e.  NN  /\  n  =  k )  ->  1  =  1 )
10 id 19 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN )
11 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
1211a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  1  e.  CC )
138, 9, 10, 12fvmptd 5622 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  1 ) `  k
)  =  1 )
1413adantl 452 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  =  1 )
151, 3, 6, 7, 14climconst 12033 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  1 )  ~~>  1 )
16 clim1fr1.1 . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
174mptex 5762 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( A  x.  n
)  +  B )  /  ( A  x.  n ) ) )  e.  _V
1816, 17eqeltri 2366 . . . 4  |-  F  e. 
_V
1918a1i 10 . . 3  |-  ( ph  ->  F  e.  _V )
20 clim1fr1.4 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2120adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  B  e.  CC )
22 clim1fr1.2 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2322adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
24 nncn 9770 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  CC )
2524adantl 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
26 clim1fr1.3 . . . . . . 7  |-  ( ph  ->  A  =/=  0 )
2726adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  =/=  0 )
28 nnne0 9794 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
2928adantl 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3021, 23, 25, 27, 29divdiv1d 9583 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( B  /  A )  /  n )  =  ( B  /  ( A  x.  n )
) )
3130mpteq2dva 4122 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  =  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) )
3220, 22, 26divcld 9552 . . . . 5  |-  ( ph  ->  ( B  /  A
)  e.  CC )
33 divcnv 12328 . . . . 5  |-  ( ( B  /  A )  e.  CC  ->  (
n  e.  NN  |->  ( ( B  /  A
)  /  n ) )  ~~>  0 )
3432, 33syl 15 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  ~~>  0 )
3531, 34eqbrtrrd 4061 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) )  ~~>  0 )
3611a1i 10 . . . . . . 7  |-  ( n  e.  NN  ->  1  e.  CC )
3736rgen 2621 . . . . . 6  |-  A. n  e.  NN  1  e.  CC
38 eqid 2296 . . . . . . 7  |-  ( n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 )
3938fmpt 5697 . . . . . 6  |-  ( A. n  e.  NN  1  e.  CC  <->  ( n  e.  NN  |->  1 ) : NN --> CC )
4037, 39mpbi 199 . . . . 5  |-  ( n  e.  NN  |->  1 ) : NN --> CC
4140a1i 10 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 ) : NN --> CC )
4241ffvelrnda 5681 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  e.  CC )
43 nfv 1609 . . . . . 6  |-  F/ n ph
4423, 25mulcld 8871 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  e.  CC )
4523, 25, 27, 29mulne0d 9436 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  =/=  0 )
4621, 44, 45divcld 9552 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( A  x.  n ) )  e.  CC )
4746ex 423 . . . . . 6  |-  ( ph  ->  ( n  e.  NN  ->  ( B  /  ( A  x.  n )
)  e.  CC ) )
4843, 47ralrimi 2637 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( B  /  ( A  x.  n )
)  e.  CC )
49 eqid 2296 . . . . . 6  |-  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) )
5049fmpt 5697 . . . . 5  |-  ( A. n  e.  NN  ( B  /  ( A  x.  n ) )  e.  CC  <->  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) : NN --> CC )
5148, 50sylib 188 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) ) : NN --> CC )
5251ffvelrnda 5681 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  e.  CC )
5316a1i 10 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) ) )
54 oveq2 5882 . . . . . . . 8  |-  ( n  =  k  ->  ( A  x.  n )  =  ( A  x.  k ) )
5554oveq1d 5889 . . . . . . 7  |-  ( n  =  k  ->  (
( A  x.  n
)  +  B )  =  ( ( A  x.  k )  +  B ) )
5655, 54oveq12d 5892 . . . . . 6  |-  ( n  =  k  ->  (
( ( A  x.  n )  +  B
)  /  ( A  x.  n ) )  =  ( ( ( A  x.  k )  +  B )  / 
( A  x.  k
) ) )
5756adantl 452 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
)  =  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) ) )
58 simpr 447 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5922adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
6058nncnd 9778 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  CC )
6159, 60mulcld 8871 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  e.  CC )
6220adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  CC )
6361, 62addcld 8870 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  +  B )  e.  CC )
6426adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  A  =/=  0 )
6558nnne0d 9806 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  =/=  0 )
6659, 60, 64, 65mulne0d 9436 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  =/=  0 )
6763, 61, 66divcld 9552 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  e.  CC )
6853, 57, 58, 67fvmptd 5622 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( A  x.  k )  +  B )  /  ( A  x.  k )
) )
6961, 62, 61, 66divdird 9590 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( ( ( A  x.  k )  / 
( A  x.  k
) )  +  ( B  /  ( A  x.  k ) ) ) )
7061, 66dividd 9550 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  /  ( A  x.  k ) )  =  1 )
7170oveq1d 5889 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  /  ( A  x.  k ) )  +  ( B  / 
( A  x.  k
) ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
7269, 71eqtrd 2328 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
7314eqcomd 2301 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  1  =  ( ( n  e.  NN  |->  1 ) `  k ) )
74 eqidd 2297 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) )
75 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  ->  n  =  k )
7675oveq2d 5890 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( A  x.  n
)  =  ( A  x.  k ) )
7776oveq2d 5890 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( B  /  ( A  x.  n )
)  =  ( B  /  ( A  x.  k ) ) )
7862, 61, 66divcld 9552 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  e.  CC )
7974, 77, 58, 78fvmptd 5622 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  =  ( B  / 
( A  x.  k
) ) )
8079eqcomd 2301 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  =  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) )
8173, 80oveq12d 5892 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  +  ( B  / 
( A  x.  k
) ) )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
8268, 72, 813eqtrd 2332 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
831, 3, 15, 19, 35, 42, 52, 82climadd 12121 . 2  |-  ( ph  ->  F  ~~>  ( 1  +  0 ) )
8411addid1i 9015 . 2  |-  ( 1  +  0 )  =  1
8583, 84syl6breq 4078 1  |-  ( ph  ->  F  ~~>  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    / cdiv 9439   NNcn 9762   ZZcz 10040    ~~> cli 11974
This theorem is referenced by:  wallispilem5  27920
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-rep 4147  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  ax-pre-sup 8831
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-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  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-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979
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