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Theorem ef0lem 13486
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 11010 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2553 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 10696 . . . . 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 10701 . . . . . . . . 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 13484 . . . . . . . 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 6210 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
12 0exp 12020 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1311, 12sylan9eq 2515 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1413oveq1d 6218 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
15 faccl 12182 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
167, 15syl 16 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ! `  k )  e.  NN )
17 nncn 10445 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
18 nnne0 10469 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
1917, 18div0d 10221 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2016, 19syl 16 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2110, 14, 203eqtrd 2499 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
22 nnne0 10469 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
23 elsn 4002 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2423necon3bbii 2713 . . . . . . . . 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 3910 . . . . . . 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 2498 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
30 fveq2 5802 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
31 oveq1 6210 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
32 0exp0e1 11991 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
3331, 32syl6eq 2511 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3433oveq1d 6218 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
35 0nn0 10709 . . . . . . . . 9  |-  0  e.  NN0
368eftval 13484 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
3735, 36ax-mp 5 . . . . . . . 8  |-  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
38 fac0 12175 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
3938oveq2i 6214 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
40 1div1e1 10139 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4139, 40eqtr2i 2484 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4234, 37, 413eqtr4g 2520 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4330, 42sylan9eqr 2517 . . . . . 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 3908 . . . . . . 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 2498 . . . . 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 2540 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5251a1i 11 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
53 1cnd 9517 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
54 0z 10772 . . . . . 6  |-  0  e.  ZZ
55 fzsn 11621 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5654, 55ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
5756eqimss2i 3522 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
5857a1i 11 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
5950, 52, 53, 58fsumcvg2 13326 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
6054, 42seq1i 11941 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
6159, 60breqtrd 4427 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648    C_ wss 3439   ifcif 3902   {csn 3988   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398    + caddc 9400    / cdiv 10108   NNcn 10437   NN0cn0 10694   ZZcz 10761   ZZ>=cuz 10976   ...cfz 11558    seqcseq 11927   ^cexp 11986   !cfa 12172    ~~> cli 13084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-seq 11928  df-exp 11987  df-fac 12173  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088
This theorem is referenced by:  ef0  13498
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