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Theorem ef0lem 12673
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
eftval.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 variable  k is distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 10512 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2526 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 10215 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 189 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 nnnn0 10220 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
76adantl 453 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
8 eftval.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
98eftval 12671 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
107, 9syl 16 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 oveq1 6080 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
12 0exp 11407 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1311, 12sylan9eq 2487 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1413oveq1d 6088 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
15 faccl 11568 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
167, 15syl 16 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ! `  k )  e.  NN )
17 nncn 10000 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
18 nnne0 10024 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
1917, 18div0d 9781 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2016, 19syl 16 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2110, 14, 203eqtrd 2471 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
22 nnne0 10024 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
23 elsn 3821 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2423necon3bbii 2629 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2522, 24sylibr 204 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2625adantl 453 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
27 iffalse 3738 . . . . . . 7  |-  ( -.  k  e.  { 0 }  ->  if (
k  e.  { 0 } ,  1 ,  0 )  =  0 )
2826, 27syl 16 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
2921, 28eqtr4d 2470 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
30 fveq2 5720 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
31 oveq1 6080 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
32 0cn 9076 . . . . . . . . . . 11  |-  0  e.  CC
33 exp0 11378 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
3432, 33ax-mp 8 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
3531, 34syl6eq 2483 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3635oveq1d 6088 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
37 0nn0 10228 . . . . . . . . 9  |-  0  e.  NN0
388eftval 12671 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
40 fac0 11561 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 6084 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 ax-1cn 9040 . . . . . . . . . 10  |-  1  e.  CC
4342div1i 9734 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4441, 43eqtr2i 2456 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4536, 39, 443eqtr4g 2492 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4630, 45sylan9eqr 2489 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
47 simpr 448 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4847, 23sylibr 204 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
49 iftrue 3737 . . . . . . 7  |-  ( k  e.  { 0 }  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
5048, 49syl 16 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
5146, 50eqtr4d 2470 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5229, 51jaodan 761 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
535, 52syldan 457 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5437, 2eleqtri 2507 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5554a1i 11 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
5642a1i 11 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
57 0z 10285 . . . . . 6  |-  0  e.  ZZ
58 fzsn 11086 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5957, 58ax-mp 8 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6059eqimss2i 3395 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6160a1i 11 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6253, 55, 56, 61fsumcvg2 12513 . 2  |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  (  seq  0 (  +  ,  F ) `  0
) )
6357, 45seq1i 11329 . 2  |-  ( A  =  0  ->  (  seq  0 (  +  ,  F ) `  0
)  =  1 )
6462, 63breqtrd 4228 1  |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   ^cexp 11374   !cfa 11558    ~~> cli 12270
This theorem is referenced by:  ef0  12685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-fac 11559  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274
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