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Theorem binomcxplemfrat 36770
Description: Lemma for binomcxp 36776. binomcxplemrat 36769 implies that when  C is not a nonnegative integer, the absolute value of the ratio  ( ( F `
 ( k  +  1 ) )  / 
( F `  k
) ) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
Hypotheses
Ref Expression
binomcxp.a  |-  ( ph  ->  A  e.  RR+ )
binomcxp.b  |-  ( ph  ->  B  e.  RR )
binomcxp.lt  |-  ( ph  ->  ( abs `  B
)  <  ( abs `  A ) )
binomcxp.c  |-  ( ph  ->  C  e.  CC )
binomcxplem.f  |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )
Assertion
Ref Expression
binomcxplemfrat  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
k  e.  NN0  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k )
) ) )  ~~>  1 )
Distinct variable groups:    j, k, ph    C, j, k
Allowed substitution hints:    A( j, k)    B( j, k)    F( j, k)

Proof of Theorem binomcxplemfrat
StepHypRef Expression
1 binomcxp.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
21adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  CC )
3 simpr 468 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
42, 3bccp1k 36760 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 (
k  +  1 ) )  =  ( ( CC𝑐 k )  x.  (
( C  -  k
)  /  ( k  +  1 ) ) ) )
5 binomcxplem.f . . . . . . . . . 10  |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )
65a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) ) )
7 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  ( k  +  1 ) )  -> 
j  =  ( k  +  1 ) )
87oveq2d 6324 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  ( k  +  1 ) )  -> 
( CC𝑐 j )  =  ( CC𝑐 ( k  +  1 ) ) )
9 1nn0 10909 . . . . . . . . . . 11  |-  1  e.  NN0
109a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  e.  NN0 )
113, 10nn0addcld 10953 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  +  1 )  e. 
NN0 )
12 ovex 6336 . . . . . . . . . 10  |-  ( CC𝑐 ( k  +  1 ) )  e.  _V
1312a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 (
k  +  1 ) )  e.  _V )
146, 8, 11, 13fvmptd 5969 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  ( k  +  1 ) )  =  ( CC𝑐 ( k  +  1 ) ) )
15 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
j  =  k )
1615oveq2d 6324 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
( CC𝑐 j )  =  ( CC𝑐 k ) )
17 ovex 6336 . . . . . . . . . . 11  |-  ( CC𝑐 k )  e.  _V
1817a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 k
)  e.  _V )
196, 16, 3, 18fvmptd 5969 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( CC𝑐 k ) )
2019oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( F `  k )  x.  ( ( C  -  k )  /  (
k  +  1 ) ) )  =  ( ( CC𝑐 k )  x.  (
( C  -  k
)  /  ( k  +  1 ) ) ) )
214, 14, 203eqtr4d 2515 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  ( k  +  1 ) )  =  ( ( F `  k
)  x.  ( ( C  -  k )  /  ( k  +  1 ) ) ) )
2221adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( F `  (
k  +  1 ) )  =  ( ( F `  k )  x.  ( ( C  -  k )  / 
( k  +  1 ) ) ) )
2322eqcomd 2477 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( F `  k )  x.  (
( C  -  k
)  /  ( k  +  1 ) ) )  =  ( F `
 ( k  +  1 ) ) )
242, 3bcccl 36758 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 k
)  e.  CC )
2519, 24eqeltrd 2549 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
2625adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( F `  k
)  e.  CC )
272adantlr 729 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  ->  C  e.  CC )
28 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
k  e.  NN0 )
2928nn0cnd 10951 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
k  e.  CC )
3027, 29subcld 10005 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( C  -  k
)  e.  CC )
31 1cnd 9677 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
1  e.  CC )
3229, 31addcld 9680 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( k  +  1 )  e.  CC )
33 nn0p1nn 10933 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3433nnne0d 10676 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( k  +  1 )  =/=  0 )
3534adantl 473 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( k  +  1 )  =/=  0 )
3630, 32, 35divcld 10405 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( C  -  k )  /  (
k  +  1 ) )  e.  CC )
3726, 36mulcld 9681 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( F `  k )  x.  (
( C  -  k
)  /  ( k  +  1 ) ) )  e.  CC )
3822, 37eqeltrd 2549 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( F `  (
k  +  1 ) )  e.  CC )
3919adantlr 729 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( CC𝑐 k ) )
40 elfznn0 11913 . . . . . . . . . 10  |-  ( C  e.  ( 0 ... ( k  -  1 ) )  ->  C  e.  NN0 )
4140con3i 142 . . . . . . . . 9  |-  ( -.  C  e.  NN0  ->  -.  C  e.  ( 0 ... ( k  - 
1 ) ) )
4241ad2antlr 741 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  ->  -.  C  e.  (
0 ... ( k  - 
1 ) ) )
4327, 28bcc0 36759 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( CC𝑐 k )  =  0  <->  C  e.  ( 0 ... (
k  -  1 ) ) ) )
4443necon3abid 2679 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( CC𝑐 k )  =/=  0  <->  -.  C  e.  ( 0 ... (
k  -  1 ) ) ) )
4542, 44mpbird 240 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( CC𝑐 k )  =/=  0
)
4639, 45eqnetrd 2710 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( F `  k
)  =/=  0 )
4738, 26, 36, 46divmuld 10427 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( ( F `
 ( k  +  1 ) )  / 
( F `  k
) )  =  ( ( C  -  k
)  /  ( k  +  1 ) )  <-> 
( ( F `  k )  x.  (
( C  -  k
)  /  ( k  +  1 ) ) )  =  ( F `
 ( k  +  1 ) ) ) )
4823, 47mpbird 240 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( ( F `  ( k  +  1 ) )  /  ( F `  k )
)  =  ( ( C  -  k )  /  ( k  +  1 ) ) )
4948fveq2d 5883 . . 3  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  k  e.  NN0 )  -> 
( abs `  (
( F `  (
k  +  1 ) )  /  ( F `
 k ) ) )  =  ( abs `  ( ( C  -  k )  /  (
k  +  1 ) ) ) )
5049mpteq2dva 4482 . 2  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
k  e.  NN0  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k )
) ) )  =  ( k  e.  NN0  |->  ( abs `  ( ( C  -  k )  /  ( k  +  1 ) ) ) ) )
51 binomcxp.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
52 binomcxp.b . . . 4  |-  ( ph  ->  B  e.  RR )
53 binomcxp.lt . . . 4  |-  ( ph  ->  ( abs `  B
)  <  ( abs `  A ) )
5451, 52, 53, 1binomcxplemrat 36769 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  ( abs `  ( ( C  -  k )  /  ( k  +  1 ) ) ) )  ~~>  1 )
5554adantr 472 . 2  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
k  e.  NN0  |->  ( abs `  ( ( C  -  k )  /  (
k  +  1 ) ) ) )  ~~>  1 )
5650, 55eqbrtrd 4416 1  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
k  e.  NN0  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k )
) ) )  ~~>  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880    / cdiv 10291   NN0cn0 10893   RR+crp 11325   ...cfz 11810   abscabs 13374    ~~> cli 13625  C𝑐cbcc 36755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-prod 14037  df-fallfac 14137  df-bcc 36756
This theorem is referenced by:  binomcxplemradcnv  36771
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