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Theorem ef0lem 11276
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 109 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 9312 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2209 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 8933 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 121 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 0cnd 7723 . . . . . . . . 9  |-  ( A  =  0  ->  0  e.  CC )
7 eleq1 2178 . . . . . . . . 9  |-  ( A  =  0  ->  ( A  e.  CC  <->  0  e.  CC ) )
86, 7mpbird 166 . . . . . . . 8  |-  ( A  =  0  ->  A  e.  CC )
9 nnnn0 8938 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
109adantl 273 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
11 efcllem.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
1211eftvalcn 11273 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
138, 10, 12syl2an2r 567 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
14 oveq1 5747 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
15 0exp 10279 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1614, 15sylan9eq 2168 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1716oveq1d 5755 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
18 faccl 10432 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
19 nncn 8688 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
20 nnap0 8709 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k ) #  0 )
2119, 20div0apd 8510 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
2210, 18, 213syl 17 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2313, 17, 223eqtrd 2152 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
24 nnne0 8708 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
25 velsn 3512 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2625necon3bbii 2320 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2724, 26sylibr 133 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2827adantl 273 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
2928iffalsed 3452 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
3023, 29eqtr4d 2151 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
31 fveq2 5387 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
32 0nn0 8946 . . . . . . . . . 10  |-  0  e.  NN0
3311eftvalcn 11273 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( F `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
348, 32, 33sylancl 407 . . . . . . . . 9  |-  ( A  =  0  ->  ( F `  0 )  =  ( ( A ^ 0 )  / 
( ! `  0
) ) )
35 oveq1 5747 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
36 0exp0e1 10249 . . . . . . . . . . 11  |-  ( 0 ^ 0 )  =  1
3735, 36syl6eq 2164 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3837oveq1d 5755 . . . . . . . . 9  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
3934, 38eqtrd 2148 . . . . . . . 8  |-  ( A  =  0  ->  ( F `  0 )  =  ( 1  / 
( ! `  0
) ) )
40 fac0 10425 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
4140oveq2i 5751 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
42 1div1e1 8427 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
4341, 42eqtr2i 2137 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4439, 43syl6eqr 2166 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4531, 44sylan9eqr 2170 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
46 simpr 109 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4746, 25sylibr 133 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
4847iftrued 3449 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4945, 48eqtr4d 2151 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
5030, 49jaodan 769 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
515, 50syldan 278 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
5232, 2eleqtri 2190 . . . 4  |-  0  e.  ( ZZ>= `  0 )
5352a1i 9 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
54 1cnd 7746 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
5525biimpri 132 . . . . . . 7  |-  ( k  =  0  ->  k  e.  { 0 } )
5627, 55orim12i 731 . . . . . 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 701 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  {
0 }  \/  -.  k  e.  { 0 } ) )
59 df-dc 803 . . . 4  |-  (DECID  k  e. 
{ 0 }  <->  ( k  e.  { 0 }  \/  -.  k  e.  { 0 } ) )
6058, 59sylibr 133 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> DECID  k  e.  { 0 } )
61 0z 9019 . . . . . 6  |-  0  e.  ZZ
62 fzsn 9797 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6361, 62ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
6463eqimss2i 3122 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
6564a1i 9 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
6651, 53, 54, 60, 65fsum3cvg2 11114 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
6761a1i 9 . . . 4  |-  ( A  =  0  ->  0  e.  ZZ )
688, 3, 12syl2an2r 567 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
69 eftcl 11270 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
708, 3, 69syl2an2r 567 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
7168, 70eqeltrd 2192 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  e.  CC )
72 addcl 7709 . . . . 5  |-  ( ( k  e.  CC  /\  y  e.  CC )  ->  ( k  +  y )  e.  CC )
7372adantl 273 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  CC  /\  y  e.  CC ) )  ->  ( k  +  y )  e.  CC )
7467, 71, 73seq3-1 10184 . . 3  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  ( F `
 0 ) )
7574, 44eqtrd 2148 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
7666, 75breqtrd 3922 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283    C_ wss 3039   ifcif 3442   {csn 3495   class class class wbr 3897    |-> cmpt 3957   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    + caddc 7587    / cdiv 8395   NNcn 8680   NN0cn0 8931   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741    seqcseq 10169   ^cexp 10243   !cfa 10422    ~~> cli 10998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-n0 8932  df-z 9009  df-uz 9279  df-rp 9394  df-fz 9742  df-seqfrec 10170  df-exp 10244  df-fac 10423  df-cj 10565  df-rsqrt 10721  df-abs 10722  df-clim 10999
This theorem is referenced by:  ef0  11288
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