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Theorem ef0lem 11004
Description: The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
Hypothesis
Ref Expression
efcllem.1  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
Assertion
Ref Expression
ef0lem  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Distinct variable group:    A, n
Allowed substitution hint:    F( n)

Proof of Theorem ef0lem
Dummy variables  k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 9107 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2182 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 8729 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 121 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 0cnd 7535 . . . . . . . . 9  |-  ( A  =  0  ->  0  e.  CC )
7 eleq1 2151 . . . . . . . . 9  |-  ( A  =  0  ->  ( A  e.  CC  <->  0  e.  CC ) )
86, 7mpbird 166 . . . . . . . 8  |-  ( A  =  0  ->  A  e.  CC )
9 nnnn0 8734 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
109adantl 272 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
11 efcllem.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
1211eftvalcn 11001 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
138, 10, 12syl2an2r 563 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
14 oveq1 5673 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
15 0exp 10044 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1614, 15sylan9eq 2141 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1716oveq1d 5681 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
18 faccl 10197 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
19 nncn 8484 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
20 nnap0 8505 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k ) #  0 )
2119, 20div0apd 8308 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2210, 18, 213syl 17 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2313, 17, 223eqtrd 2125 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
24 nnne0 8504 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
25 velsn 3467 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2625necon3bbii 2293 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2724, 26sylibr 133 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2827adantl 272 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
2928iffalsed 3407 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
3023, 29eqtr4d 2124 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
31 fveq2 5318 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
32 0nn0 8742 . . . . . . . . . 10  |-  0  e.  NN0
3311eftvalcn 11001 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( F `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
348, 32, 33sylancl 405 . . . . . . . . 9  |-  ( A  =  0  ->  ( F `  0 )  =  ( ( A ^ 0 )  / 
( ! `  0
) ) )
35 oveq1 5673 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
36 0exp0e1 10014 . . . . . . . . . . 11  |-  ( 0 ^ 0 )  =  1
3735, 36syl6eq 2137 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3837oveq1d 5681 . . . . . . . . 9  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
3934, 38eqtrd 2121 . . . . . . . 8  |-  ( A  =  0  ->  ( F `  0 )  =  ( 1  / 
( ! `  0
) ) )
40 fac0 10190 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 5677 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 1div1e1 8225 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4341, 42eqtr2i 2110 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4439, 43syl6eqr 2139 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4531, 44sylan9eqr 2143 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
46 simpr 109 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4746, 25sylibr 133 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
4847iftrued 3404 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4945, 48eqtr4d 2124 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5030, 49jaodan 747 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
515, 50syldan 277 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5232, 2eleqtri 2163 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5352a1i 9 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
54 1cnd 7558 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
5525biimpri 132 . . . . . . 7  |-  ( k  =  0  ->  k  e.  { 0 } )
5627, 55orim12i 712 . . . . . 6  |-  ( ( k  e.  NN  \/  k  =  0 )  ->  ( -.  k  e.  { 0 }  \/  k  e.  { 0 } ) )
575, 56syl 14 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( -.  k  e. 
{ 0 }  \/  k  e.  { 0 } ) )
5857orcomd 684 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  {
0 }  \/  -.  k  e.  { 0 } ) )
59 df-dc 782 . . . 4  |-  (DECID  k  e. 
{ 0 }  <->  ( k  e.  { 0 }  \/  -.  k  e.  { 0 } ) )
6058, 59sylibr 133 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> DECID  k  e.  { 0 } )
61 0z 8815 . . . . . 6  |-  0  e.  ZZ
62 fzsn 9534 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6361, 62ax-mp 7 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6463eqimss2i 3082 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6564a1i 9 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6651, 53, 54, 60, 65fsum3cvg2 10841 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
6761a1i 9 . . . 4  |-  ( A  =  0  ->  0  e.  ZZ )
688, 3, 12syl2an2r 563 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
69 eftcl 10998 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
708, 3, 69syl2an2r 563 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
7168, 70eqeltrd 2165 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  e.  CC )
72 addcl 7521 . . . . 5  |-  ( ( k  e.  CC  /\  y  e.  CC )  ->  ( k  +  y )  e.  CC )
7372adantl 272 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  CC  /\  y  e.  CC ) )  ->  ( k  +  y )  e.  CC )
7467, 71, 73seq3-1 9931 . . 3  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  ( F `
 0 ) )
7574, 44eqtrd 2121 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
7666, 75breqtrd 3875 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 665  DECID wdc 781    = wceq 1290    e. wcel 1439    =/= wne 2256    C_ wss 3000   ifcif 3397   {csn 3450   class class class wbr 3851    |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405    + caddc 7407    / cdiv 8193   NNcn 8476   NN0cn0 8727   ZZcz 8804   ZZ>=cuz 9073   ...cfz 9478    seqcseq 9906   ^cexp 10008   !cfa 10187    ~~> cli 10720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-n0 8728  df-z 8805  df-uz 9074  df-rp 9189  df-fz 9479  df-iseq 9907  df-seq3 9908  df-exp 10009  df-fac 10188  df-cj 10330  df-rsqrt 10485  df-abs 10486  df-clim 10721
This theorem is referenced by:  ef0  11016
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