MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ef0lem Unicode version

Theorem ef0lem 12360
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 447 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 10262 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2374 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 9967 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 188 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 nnnn0 9972 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
76adantl 452 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
8 eftval.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
98eftval 12358 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
107, 9syl 15 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 oveq1 5865 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
12 0exp 11137 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1311, 12sylan9eq 2335 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1413oveq1d 5873 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
15 faccl 11298 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
167, 15syl 15 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ! `  k )  e.  NN )
17 nncn 9754 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
18 nnne0 9778 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
1917, 18div0d 9535 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2016, 19syl 15 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2110, 14, 203eqtrd 2319 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
22 nnne0 9778 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
23 elsn 3655 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2423necon3bbii 2477 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2522, 24sylibr 203 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2625adantl 452 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
27 iffalse 3572 . . . . . . 7  |-  ( -.  k  e.  { 0 }  ->  if (
k  e.  { 0 } ,  1 ,  0 )  =  0 )
2826, 27syl 15 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
2921, 28eqtr4d 2318 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
30 fveq2 5525 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
31 oveq1 5865 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
32 0cn 8831 . . . . . . . . . . 11  |-  0  e.  CC
33 exp0 11108 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
3432, 33ax-mp 8 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
3531, 34syl6eq 2331 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3635oveq1d 5873 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
37 0nn0 9980 . . . . . . . . 9  |-  0  e.  NN0
388eftval 12358 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
40 fac0 11291 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 5869 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
4342div1i 9488 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4441, 43eqtr2i 2304 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4536, 39, 443eqtr4g 2340 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4630, 45sylan9eqr 2337 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
47 simpr 447 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4847, 23sylibr 203 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
49 iftrue 3571 . . . . . . 7  |-  ( k  e.  { 0 }  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
5048, 49syl 15 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
5146, 50eqtr4d 2318 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5229, 51jaodan 760 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
535, 52syldan 456 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5437, 2eleqtri 2355 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5554a1i 10 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
5642a1i 10 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
57 0z 10035 . . . . . 6  |-  0  e.  ZZ
58 fzsn 10833 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5957, 58ax-mp 8 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6059eqimss2i 3233 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6160a1i 10 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6253, 55, 56, 61fsumcvg2 12200 . 2  |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  (  seq  0 (  +  ,  F ) `  0
) )
6357, 45seq1i 11060 . 2  |-  ( A  =  0  ->  (  seq  0 (  +  ,  F ) `  0
)  =  1 )
6462, 63breqtrd 4047 1  |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   ^cexp 11104   !cfa 11288    ~~> cli 11958
This theorem is referenced by:  ef0  12372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-fac 11289  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
  Copyright terms: Public domain W3C validator