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Theorem binomcxplemradcnv 36771
Description: Lemma for binomcxp 36776. By binomcxplemfrat 36770 and radcnvrat 36733 the radius of convergence of power series  sum_ k  e.  NN0 ( ( F `  k )  x.  (
b ^ k ) ) is one. (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 ) )
binomcxplem.s  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
binomcxplem.r  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
Assertion
Ref Expression
binomcxplemradcnv  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
Distinct variable groups:    C, k    k, b, F    j, k, ph    C, j    S, r
Allowed substitution hints:    ph( r, b)    A( j, k, r, b)    B( j, k, r, b)    C( r, b)    R( j, k, r, b)    S( j, k, b)    F( j, r)

Proof of Theorem binomcxplemradcnv
Dummy variables  i  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 binomcxplem.s . . . 4  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
2 simpl 464 . . . . . . . . 9  |-  ( ( b  =  x  /\  k  e.  NN0 )  -> 
b  =  x )
32oveq1d 6323 . . . . . . . 8  |-  ( ( b  =  x  /\  k  e.  NN0 )  -> 
( b ^ k
)  =  ( x ^ k ) )
43oveq2d 6324 . . . . . . 7  |-  ( ( b  =  x  /\  k  e.  NN0 )  -> 
( ( F `  k )  x.  (
b ^ k ) )  =  ( ( F `  k )  x.  ( x ^
k ) ) )
54mpteq2dva 4482 . . . . . 6  |-  ( b  =  x  ->  (
k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) )  =  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( x ^ k
) ) ) )
6 fveq2 5879 . . . . . . . 8  |-  ( k  =  y  ->  ( F `  k )  =  ( F `  y ) )
7 oveq2 6316 . . . . . . . 8  |-  ( k  =  y  ->  (
x ^ k )  =  ( x ^
y ) )
86, 7oveq12d 6326 . . . . . . 7  |-  ( k  =  y  ->  (
( F `  k
)  x.  ( x ^ k ) )  =  ( ( F `
 y )  x.  ( x ^ y
) ) )
98cbvmptv 4488 . . . . . 6  |-  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( x ^
k ) ) )  =  ( y  e. 
NN0  |->  ( ( F `
 y )  x.  ( x ^ y
) ) )
105, 9syl6eq 2521 . . . . 5  |-  ( b  =  x  ->  (
k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) )  =  ( y  e. 
NN0  |->  ( ( F `
 y )  x.  ( x ^ y
) ) ) )
1110cbvmptv 4488 . . . 4  |-  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) )  =  ( x  e.  CC  |->  ( y  e.  NN0  |->  ( ( F `  y )  x.  ( x ^
y ) ) ) )
121, 11eqtri 2493 . . 3  |-  S  =  ( x  e.  CC  |->  ( y  e.  NN0  |->  ( ( F `  y )  x.  (
x ^ y ) ) ) )
13 binomcxp.c . . . . . 6  |-  ( ph  ->  C  e.  CC )
1413ad2antrr 740 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  j  e.  NN0 )  ->  C  e.  CC )
15 simpr 468 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  j  e.  NN0 )  -> 
j  e.  NN0 )
1614, 15bcccl 36758 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  j  e.  NN0 )  -> 
( CC𝑐 j )  e.  CC )
17 binomcxplem.f . . . 4  |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )
1816, 17fmptd 6061 . . 3  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  F : NN0 --> CC )
19 binomcxplem.r . . 3  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
20 oveq1 6315 . . . . . . 7  |-  ( k  =  i  ->  (
k  +  1 )  =  ( i  +  1 ) )
2120fveq2d 5883 . . . . . 6  |-  ( k  =  i  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( i  +  1 ) ) )
22 fveq2 5879 . . . . . 6  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
2321, 22oveq12d 6326 . . . . 5  |-  ( k  =  i  ->  (
( F `  (
k  +  1 ) )  /  ( F `
 k ) )  =  ( ( F `
 ( i  +  1 ) )  / 
( F `  i
) ) )
2423fveq2d 5883 . . . 4  |-  ( k  =  i  ->  ( abs `  ( ( F `
 ( k  +  1 ) )  / 
( F `  k
) ) )  =  ( abs `  (
( F `  (
i  +  1 ) )  /  ( F `
 i ) ) ) )
2524cbvmptv 4488 . . 3  |-  ( k  e.  NN0  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k )
) ) )  =  ( i  e.  NN0  |->  ( abs `  ( ( F `  ( i  +  1 ) )  /  ( F `  i ) ) ) )
26 nn0uz 11217 . . 3  |-  NN0  =  ( ZZ>= `  0 )
27 0nn0 10908 . . . 4  |-  0  e.  NN0
2827a1i 11 . . 3  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  0  e.  NN0 )
2917a1i 11 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  ->  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) ) )
30 simpr 468 . . . . . 6  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  /\  j  =  i )  ->  j  =  i )
3130oveq2d 6324 . . . . 5  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  /\  j  =  i )  ->  ( CC𝑐 j
)  =  ( CC𝑐 i ) )
32 simpr 468 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
i  e.  NN0 )
33 ovex 6336 . . . . . 6  |-  ( CC𝑐 i )  e.  _V
3433a1i 11 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
( CC𝑐 i )  e.  _V )
3529, 31, 32, 34fvmptd 5969 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
( F `  i
)  =  ( CC𝑐 i ) )
36 elfznn0 11913 . . . . . . 7  |-  ( C  e.  ( 0 ... ( i  -  1 ) )  ->  C  e.  NN0 )
3736con3i 142 . . . . . 6  |-  ( -.  C  e.  NN0  ->  -.  C  e.  ( 0 ... ( i  - 
1 ) ) )
3837ad2antlr 741 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  ->  -.  C  e.  (
0 ... ( i  - 
1 ) ) )
3913adantr 472 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN0 )  ->  C  e.  CC )
40 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN0 )  ->  i  e.  NN0 )
4139, 40bcc0 36759 . . . . . . 7  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( ( CC𝑐 i )  =  0  <-> 
C  e.  ( 0 ... ( i  - 
1 ) ) ) )
4241necon3abid 2679 . . . . . 6  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( ( CC𝑐 i )  =/=  0  <->  -.  C  e.  ( 0 ... ( i  - 
1 ) ) ) )
4342adantlr 729 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
( ( CC𝑐 i )  =/=  0  <->  -.  C  e.  ( 0 ... (
i  -  1 ) ) ) )
4438, 43mpbird 240 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
( CC𝑐 i )  =/=  0
)
4535, 44eqnetrd 2710 . . 3  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  i  e.  NN0 )  -> 
( F `  i
)  =/=  0 )
46 binomcxp.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
47 binomcxp.b . . . 4  |-  ( ph  ->  B  e.  RR )
48 binomcxp.lt . . . 4  |-  ( ph  ->  ( abs `  B
)  <  ( abs `  A ) )
4946, 47, 48, 13, 17binomcxplemfrat 36770 . . 3  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
k  e.  NN0  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k )
) ) )  ~~>  1 )
50 ax-1ne0 9626 . . . 4  |-  1  =/=  0
5150a1i 11 . . 3  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  1  =/=  0 )
5212, 18, 19, 25, 26, 28, 45, 49, 51radcnvrat 36733 . 2  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  ( 1  / 
1 ) )
53 1div1e1 10322 . 2  |-  ( 1  /  1 )  =  1
5452, 53syl6eq 2521 1  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    - cmin 9880    / cdiv 10291   NN0cn0 10893   RR+crp 11325   ...cfz 11810    seqcseq 12251   ^cexp 12310   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  ax-addf 9636  ax-mulf 9637
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-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  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-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-prod 14037  df-fallfac 14137  df-bcc 36756
This theorem is referenced by:  binomcxplemdvbinom  36772  binomcxplemnotnn0  36775
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