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Theorem ef0lem 11971
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 9683 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtrrdi 2299 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 9297 . . . . 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 8065 . . . . . . . . 9  |-  ( A  =  0  ->  0  e.  CC )
7 eleq1 2268 . . . . . . . . 9  |-  ( A  =  0  ->  ( A  e.  CC  <->  0  e.  CC ) )
86, 7mpbird 167 . . . . . . . 8  |-  ( A  =  0  ->  A  e.  CC )
9 nnnn0 9302 . . . . . . . . 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 11968 . . . . . . . 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 5951 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
15 0exp 10719 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1614, 15sylan9eq 2258 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1716oveq1d 5959 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
18 faccl 10880 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
19 nncn 9044 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
20 nnap0 9065 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k ) #  0 )
2119, 20div0apd 8860 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2210, 18, 213syl 17 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2313, 17, 223eqtrd 2242 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
24 nnne0 9064 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
25 velsn 3650 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2625necon3bbii 2413 . . . . . . . . 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 3581 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
3023, 29eqtr4d 2241 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
31 fveq2 5576 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
32 0nn0 9310 . . . . . . . . . 10  |-  0  e.  NN0
3311eftvalcn 11968 . . . . . . . . . 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 5951 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
36 0exp0e1 10689 . . . . . . . . . . 11  |-  ( 0 ^ 0 )  =  1
3735, 36eqtrdi 2254 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3837oveq1d 5959 . . . . . . . . 9  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
3934, 38eqtrd 2238 . . . . . . . 8  |-  ( A  =  0  ->  ( F `  0 )  =  ( 1  / 
( ! `  0
) ) )
40 fac0 10873 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 5955 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 1div1e1 8777 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4341, 42eqtr2i 2227 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4439, 43eqtr4di 2256 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4531, 44sylan9eqr 2260 . . . . . 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 3578 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4945, 48eqtr4d 2241 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5030, 49jaodan 799 . . . 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 2280 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5352a1i 9 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
54 1cnd 8088 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
5525biimpri 133 . . . . . . 7  |-  ( k  =  0  ->  k  e.  { 0 } )
5627, 55orim12i 761 . . . . . 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 731 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  {
0 }  \/  -.  k  e.  { 0 } ) )
59 df-dc 837 . . . 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 9383 . . . . . 6  |-  0  e.  ZZ
62 fzsn 10188 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6361, 62ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6463eqimss2i 3250 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6564a1i 9 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6651, 53, 54, 60, 65fsum3cvg2 11705 . 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 11965 . . . . . 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 2282 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  e.  CC )
72 addcl 8050 . . . . 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 10607 . . 3  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  ( F `
 0 ) )
7574, 44eqtrd 2238 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
7666, 75breqtrd 4070 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 710  DECID wdc 836    = wceq 1373    e. wcel 2176    =/= wne 2376    C_ wss 3166   ifcif 3571   {csn 3633   class class class wbr 4044    |-> cmpt 4105   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926    + caddc 7928    / cdiv 8745   NNcn 9036   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130    seqcseq 10592   ^cexp 10683   !cfa 10870    ~~> cli 11589
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-fz 10131  df-seqfrec 10593  df-exp 10684  df-fac 10871  df-cj 11153  df-rsqrt 11309  df-abs 11310  df-clim 11590
This theorem is referenced by:  ef0  11983
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