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Theorem ef0lem 13688
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 461 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 11126 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2566 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 10807 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 196 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 nnnn0 10812 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
76adantl 466 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
8 eftval.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
98eftval 13686 . . . . . . . 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 6301 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
12 0exp 12179 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1311, 12sylan9eq 2528 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1413oveq1d 6309 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
15 faccl 12341 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
167, 15syl 16 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ! `  k )  e.  NN )
17 nncn 10554 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
18 nnne0 10578 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
1917, 18div0d 10329 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2016, 19syl 16 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2110, 14, 203eqtrd 2512 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
22 nnne0 10578 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
23 elsn 4046 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2423necon3bbii 2728 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2522, 24sylibr 212 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2625adantl 466 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
27 iffalse 3953 . . . . . . 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 2511 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
30 fveq2 5871 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
31 oveq1 6301 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
32 0exp0e1 12149 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
3331, 32syl6eq 2524 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3433oveq1d 6309 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
35 0nn0 10820 . . . . . . . . 9  |-  0  e.  NN0
368eftval 13686 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
3735, 36ax-mp 5 . . . . . . . 8  |-  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
38 fac0 12334 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
3938oveq2i 6305 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
40 1div1e1 10247 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4139, 40eqtr2i 2497 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4234, 37, 413eqtr4g 2533 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4330, 42sylan9eqr 2530 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
44 simpr 461 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4544, 23sylibr 212 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
46 iftrue 3950 . . . . . . 7  |-  ( k  e.  { 0 }  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4745, 46syl 16 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4843, 47eqtr4d 2511 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
4929, 48jaodan 783 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
505, 49syldan 470 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5135, 2eleqtri 2553 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5251a1i 11 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
53 1cnd 9622 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
54 0z 10885 . . . . . 6  |-  0  e.  ZZ
55 fzsn 11735 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5654, 55ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
5756eqimss2i 3564 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
5857a1i 11 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
5950, 52, 53, 58fsumcvg2 13524 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
6054, 42seq1i 12099 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
6159, 60breqtrd 4476 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   ifcif 3944   {csn 4032   class class class wbr 4452    |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   0cc0 9502   1c1 9503    + caddc 9505    / cdiv 10216   NNcn 10546   NN0cn0 10805   ZZcz 10874   ZZ>=cuz 11092   ...cfz 11682    seqcseq 12085   ^cexp 12144   !cfa 12331    ~~> cli 13282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-seq 12086  df-exp 12145  df-fac 12332  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286
This theorem is referenced by:  ef0  13700
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