Theorem ef0lem 13688

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.

Step	Hyp	Ref	Expression
1		simpr 461	. . . . . 6 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. ( ZZ>= ` 0 ) )
2		nn0uz 11126	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleqr 2566	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 10807	. . . . 5 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
5	3, 4	sylib 196	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. NN \/ k = 0 ) )
6		nnnn0 10812	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
7	6	adantl 466	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> k e. NN0 )
8		eftval.1	. . . . . . . . 9 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
9	8	eftval 13686	. . . . . . . 8 |- ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
10	7, 9	syl 16	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
11		oveq1 6301	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
12		0exp 12179	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
13	11, 12	sylan9eq 2528	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
14	13	oveq1d 6309	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
15		faccl 12341	. . . . . . . . 9 |- ( k e. NN0 -> ( ! ` k ) e. NN )
16	7, 15	syl 16	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( ! ` k ) e. NN )
17		nncn 10554	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
18		nnne0 10578	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
19	17, 18	div0d 10329	. . . . . . . 8 |- ( ( ! ` k ) e. NN -> ( 0 / ( ! ` k ) ) = 0 )
20	16, 19	syl 16	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( 0 / ( ! ` k ) ) = 0 )
21	10, 14, 20	3eqtrd 2512	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
22		nnne0 10578	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
23		elsn 4046	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
24	23	necon3bbii 2728	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
25	22, 24	sylibr 212	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
26	25	adantl 466	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
27		iffalse 3953	. . . . . . 7 |- ( -. k e. { 0 } -> if ( k e. { 0 } , 1 , 0 ) = 0 )
28	26, 27	syl 16	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
29	21, 28	eqtr4d 2511	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
30		fveq2 5871	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
31		oveq1 6301	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
32		0exp0e1 12149	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
33	31, 32	syl6eq 2524	. . . . . . . . 9 |- ( A = 0 -> ( A ^ 0 ) = 1 )
34	33	oveq1d 6309	. . . . . . . 8 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
35		0nn0 10820	. . . . . . . . 9 |- 0 e. NN0
36	8	eftval 13686	. . . . . . . . 9 |- ( 0 e. NN0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
37	35, 36	ax-mp 5	. . . . . . . 8 |- ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) )
38		fac0 12334	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
39	38	oveq2i 6305	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
40		1div1e1 10247	. . . . . . . . 9 |- ( 1 / 1 ) = 1
41	39, 40	eqtr2i 2497	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
42	34, 37, 41	3eqtr4g 2533	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
43	30, 42	sylan9eqr 2530	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
44		simpr 461	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
45	44, 23	sylibr 212	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
46		iftrue 3950	. . . . . . 7 |- ( k e. { 0 } -> if ( k e. { 0 } , 1 , 0 ) = 1 )
47	45, 46	syl 16	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
48	43, 47	eqtr4d 2511	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
49	29, 48	jaodan 783	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	5, 49	syldan 470	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
51	35, 2	eleqtri 2553	. . . 4 |- 0 e. ( ZZ>= ` 0 )
52	51	a1i 11	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
53		1cnd 9622	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
54		0z 10885	. . . . . 6 |- 0 e. ZZ
55		fzsn 11735	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
56	54, 55	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
57	56	eqimss2i 3564	. . . 4 |- { 0 } C_ ( 0 ... 0 )
58	57	a1i 11	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
59	50, 52, 53, 58	fsumcvg2 13524	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
60	54, 42	seq1i 12099	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
61	59, 60	breqtrd 4476	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   C_ wss 3481   ifcif 3944   {csn 4032   class class class wbr 4452   |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   0cc0 9502   1c1 9503   + caddc 9505   / cdiv 10216   NNcn 10546   NN0cn0 10805   ZZcz 10874   ZZ>=cuz 11092   ...cfz 11682   seqcseq 12085   ^cexp 12144   !cfa 12331   ~~> cli 13282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-seq 12086  df-exp 12145  df-fac 12332  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286

This theorem is referenced by:  ef0  13700