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Theorem ef0lem 11942
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 110 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 9682 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtrrdi 2298 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 9296 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 122 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 0cnd 8064 . . . . . . . . 9  |-  ( A  =  0  ->  0  e.  CC )
7 eleq1 2267 . . . . . . . . 9  |-  ( A  =  0  ->  ( A  e.  CC  <->  0  e.  CC ) )
86, 7mpbird 167 . . . . . . . 8  |-  ( A  =  0  ->  A  e.  CC )
9 nnnn0 9301 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
109adantl 277 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
11 efcllem.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
1211eftvalcn 11939 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
138, 10, 12syl2an2r 595 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
14 oveq1 5950 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
15 0exp 10717 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1614, 15sylan9eq 2257 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1716oveq1d 5958 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
18 faccl 10878 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
19 nncn 9043 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
20 nnap0 9064 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k ) #  0 )
2119, 20div0apd 8859 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2210, 18, 213syl 17 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2313, 17, 223eqtrd 2241 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
24 nnne0 9063 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
25 velsn 3649 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2625necon3bbii 2412 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2724, 26sylibr 134 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2827adantl 277 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
2928iffalsed 3580 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
3023, 29eqtr4d 2240 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
31 fveq2 5575 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
32 0nn0 9309 . . . . . . . . . 10  |-  0  e.  NN0
3311eftvalcn 11939 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( F `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
348, 32, 33sylancl 413 . . . . . . . . 9  |-  ( A  =  0  ->  ( F `  0 )  =  ( ( A ^ 0 )  / 
( ! `  0
) ) )
35 oveq1 5950 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
36 0exp0e1 10687 . . . . . . . . . . 11  |-  ( 0 ^ 0 )  =  1
3735, 36eqtrdi 2253 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3837oveq1d 5958 . . . . . . . . 9  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
3934, 38eqtrd 2237 . . . . . . . 8  |-  ( A  =  0  ->  ( F `  0 )  =  ( 1  / 
( ! `  0
) ) )
40 fac0 10871 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 5954 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 1div1e1 8776 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4341, 42eqtr2i 2226 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4439, 43eqtr4di 2255 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4531, 44sylan9eqr 2259 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
46 simpr 110 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4746, 25sylibr 134 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
4847iftrued 3577 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4945, 48eqtr4d 2240 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5030, 49jaodan 798 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
515, 50syldan 282 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5232, 2eleqtri 2279 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5352a1i 9 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
54 1cnd 8087 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
5525biimpri 133 . . . . . . 7  |-  ( k  =  0  ->  k  e.  { 0 } )
5627, 55orim12i 760 . . . . . 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 730 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  {
0 }  \/  -.  k  e.  { 0 } ) )
59 df-dc 836 . . . 4  |-  (DECID  k  e. 
{ 0 }  <->  ( k  e.  { 0 }  \/  -.  k  e.  { 0 } ) )
6058, 59sylibr 134 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> DECID  k  e.  { 0 } )
61 0z 9382 . . . . . 6  |-  0  e.  ZZ
62 fzsn 10187 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6361, 62ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6463eqimss2i 3249 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6564a1i 9 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6651, 53, 54, 60, 65fsum3cvg2 11676 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
6761a1i 9 . . . 4  |-  ( A  =  0  ->  0  e.  ZZ )
688, 3, 12syl2an2r 595 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
69 eftcl 11936 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
708, 3, 69syl2an2r 595 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
7168, 70eqeltrd 2281 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  e.  CC )
72 addcl 8049 . . . . 5  |-  ( ( k  e.  CC  /\  y  e.  CC )  ->  ( k  +  y )  e.  CC )
7372adantl 277 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  CC  /\  y  e.  CC ) )  ->  ( k  +  y )  e.  CC )
7467, 71, 73seq3-1 10605 . . 3  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  ( F `
 0 ) )
7574, 44eqtrd 2237 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
7666, 75breqtrd 4069 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    = wceq 1372    e. wcel 2175    =/= wne 2375    C_ wss 3165   ifcif 3570   {csn 3632   class class class wbr 4043    |-> cmpt 4104   ` cfv 5270  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925    + caddc 7927    / cdiv 8744   NNcn 9035   NN0cn0 9294   ZZcz 9371   ZZ>=cuz 9647   ...cfz 10129    seqcseq 10590   ^cexp 10681   !cfa 10868    ~~> cli 11560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-fz 10130  df-seqfrec 10591  df-exp 10682  df-fac 10869  df-cj 11124  df-rsqrt 11280  df-abs 11281  df-clim 11561
This theorem is referenced by:  ef0  11954
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