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Theorem ef0lem 14133
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 462 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  ( ZZ>= ` 
0 ) )
2 nn0uz 11201 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleqr 2518 . . . . 5  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
k  e.  NN0 )
4 elnn0 10879 . . . . 5  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
53, 4sylib 199 . . . 4  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( k  e.  NN  \/  k  =  0
) )
6 nnnn0 10884 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
76adantl 467 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  k  e.  NN0 )
8 eftval.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
98eftval 14131 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
107, 9syl 17 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 oveq1 6313 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ k )  =  ( 0 ^ k
) )
12 0exp 12314 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1311, 12sylan9eq 2483 . . . . . . . 8  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( A ^
k )  =  0 )
1413oveq1d 6321 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( ( A ^ k )  / 
( ! `  k
) )  =  ( 0  /  ( ! `
 k ) ) )
15 faccl 12476 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
16 nncn 10625 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
17 nnne0 10650 . . . . . . . . 9  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
1816, 17div0d 10390 . . . . . . . 8  |-  ( ( ! `  k )  e.  NN  ->  (
0  /  ( ! `
 k ) )  =  0 )
197, 15, 183syl 18 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( 0  / 
( ! `  k
) )  =  0 )
2010, 14, 193eqtrd 2467 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  0 )
21 nnne0 10650 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  =/=  0 )
22 elsn 4012 . . . . . . . . . 10  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2322necon3bbii 2681 . . . . . . . . 9  |-  ( -.  k  e.  { 0 }  <->  k  =/=  0
)
2421, 23sylibr 215 . . . . . . . 8  |-  ( k  e.  NN  ->  -.  k  e.  { 0 } )
2524adantl 467 . . . . . . 7  |-  ( ( A  =  0  /\  k  e.  NN )  ->  -.  k  e.  { 0 } )
2625iffalsed 3922 . . . . . 6  |-  ( ( A  =  0  /\  k  e.  NN )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  0 )
2720, 26eqtr4d 2466 . . . . 5  |-  ( ( A  =  0  /\  k  e.  NN )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
28 fveq2 5882 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
29 oveq1 6313 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ 0 )  =  ( 0 ^ 0 ) )
30 0exp0e1 12284 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
3129, 30syl6eq 2479 . . . . . . . . 9  |-  ( A  =  0  ->  ( A ^ 0 )  =  1 )
3231oveq1d 6321 . . . . . . . 8  |-  ( A  =  0  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  ( 1  / 
( ! `  0
) ) )
33 0nn0 10892 . . . . . . . . 9  |-  0  e.  NN0
348eftval 14131 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
3533, 34ax-mp 5 . . . . . . . 8  |-  ( F `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
36 fac0 12469 . . . . . . . . . 10  |-  ( ! `
 0 )  =  1
3736oveq2i 6317 . . . . . . . . 9  |-  ( 1  /  ( ! ` 
0 ) )  =  ( 1  /  1
)
38 1div1e1 10308 . . . . . . . . 9  |-  ( 1  /  1 )  =  1
3937, 38eqtr2i 2452 . . . . . . . 8  |-  1  =  ( 1  / 
( ! `  0
) )
4032, 35, 393eqtr4g 2488 . . . . . . 7  |-  ( A  =  0  ->  ( F `  0 )  =  1 )
4128, 40sylan9eqr 2485 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  1 )
42 simpr 462 . . . . . . . 8  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  =  0 )
4342, 22sylibr 215 . . . . . . 7  |-  ( ( A  =  0  /\  k  =  0 )  ->  k  e.  {
0 } )
4443iftrued 3919 . . . . . 6  |-  ( ( A  =  0  /\  k  =  0 )  ->  if ( k  e.  { 0 } ,  1 ,  0 )  =  1 )
4541, 44eqtr4d 2466 . . . . 5  |-  ( ( A  =  0  /\  k  =  0 )  ->  ( F `  k )  =  if ( k  e.  {
0 } ,  1 ,  0 ) )
4627, 45jaodan 792 . . . 4  |-  ( ( A  =  0  /\  ( k  e.  NN  \/  k  =  0
) )  ->  ( F `  k )  =  if ( k  e. 
{ 0 } , 
1 ,  0 ) )
475, 46syldan 472 . . 3  |-  ( ( A  =  0  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( F `  k
)  =  if ( k  e.  { 0 } ,  1 ,  0 ) )
4833, 2eleqtri 2505 . . . 4  |-  0  e.  ( ZZ>= `  0 )
4948a1i 11 . . 3  |-  ( A  =  0  ->  0  e.  ( ZZ>= `  0 )
)
50 1cnd 9667 . . 3  |-  ( ( A  =  0  /\  k  e.  { 0 } )  ->  1  e.  CC )
51 0z 10956 . . . . . 6  |-  0  e.  ZZ
52 fzsn 11848 . . . . . 6  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5351, 52ax-mp 5 . . . . 5  |-  ( 0 ... 0 )  =  { 0 }
5453eqimss2i 3519 . . . 4  |-  { 0 }  C_  ( 0 ... 0 )
5554a1i 11 . . 3  |-  ( A  =  0  ->  { 0 }  C_  ( 0 ... 0 ) )
5647, 49, 50, 55fsumcvg2 13793 . 2  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  (  seq 0 (  +  ,  F ) `  0
) )
5751, 40seq1i 12234 . 2  |-  ( A  =  0  ->  (  seq 0 (  +  ,  F ) `  0
)  =  1 )
5856, 57breqtrd 4448 1  |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614    C_ wss 3436   ifcif 3911   {csn 3998   class class class wbr 4423    |-> cmpt 4482   ` cfv 5601  (class class class)co 6306   0cc0 9547   1c1 9548    + caddc 9550    / cdiv 10277   NNcn 10617   NN0cn0 10877   ZZcz 10945   ZZ>=cuz 11167   ...cfz 11792    seqcseq 12220   ^cexp 12279   !cfa 12466    ~~> cli 13548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-n0 10878  df-z 10946  df-uz 11168  df-rp 11311  df-fz 11793  df-seq 12221  df-exp 12280  df-fac 12467  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-clim 13552
This theorem is referenced by:  ef0  14145
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