Theorem ef0lem 13692

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 11124	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleqr 2542	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 10803	. . . . 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 10808	. . . . . . . . 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 13690	. . . . . . . 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 6288	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
12		0exp 12180	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
13	11, 12	sylan9eq 2504	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
14	13	oveq1d 6296	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
15		faccl 12342	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
16		nncn 10550	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
17		nnne0 10574	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
18	16, 17	div0d 10325	. . . . . . . 8 |- ( ( ! ` k ) e. NN -> ( 0 / ( ! ` k ) ) = 0 )
19	7, 15, 18	3syl 20	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( 0 / ( ! ` k ) ) = 0 )
20	10, 14, 19	3eqtrd 2488	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
21		nnne0 10574	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
22		elsn 4028	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
23	22	necon3bbii 2704	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
24	21, 23	sylibr 212	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
25	24	adantl 466	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
26	25	iffalsed 3937	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
27	20, 26	eqtr4d 2487	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
28		fveq2 5856	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
29		oveq1 6288	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
30		0exp0e1 12150	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
31	29, 30	syl6eq 2500	. . . . . . . . 9 |- ( A = 0 -> ( A ^ 0 ) = 1 )
32	31	oveq1d 6296	. . . . . . . 8 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
33		0nn0 10816	. . . . . . . . 9 |- 0 e. NN0
34	8	eftval 13690	. . . . . . . . 9 |- ( 0 e. NN0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35	33, 34	ax-mp 5	. . . . . . . 8 |- ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) )
36		fac0 12335	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
37	36	oveq2i 6292	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
38		1div1e1 10243	. . . . . . . . 9 |- ( 1 / 1 ) = 1
39	37, 38	eqtr2i 2473	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
40	32, 35, 39	3eqtr4g 2509	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
41	28, 40	sylan9eqr 2506	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
42		simpr 461	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
43	42, 22	sylibr 212	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
44	43	iftrued 3934	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
45	41, 44	eqtr4d 2487	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
46	27, 45	jaodan 785	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
47	5, 46	syldan 470	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
48	33, 2	eleqtri 2529	. . . 4 |- 0 e. ( ZZ>= ` 0 )
49	48	a1i 11	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
50		1cnd 9615	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
51		0z 10881	. . . . . 6 |- 0 e. ZZ
52		fzsn 11734	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
53	51, 52	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
54	53	eqimss2i 3544	. . . 4 |- { 0 } C_ ( 0 ... 0 )
55	54	a1i 11	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
56	47, 49, 50, 55	fsumcvg2 13528	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
57	51, 40	seq1i 12100	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
58	56, 57	breqtrd 4461	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   C_ wss 3461   ifcif 3926   {csn 4014   class class class wbr 4437   |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496   + caddc 9498   / cdiv 10212   NNcn 10542   NN0cn0 10801   ZZcz 10870   ZZ>=cuz 11090   ...cfz 11681   seqcseq 12086   ^cexp 12145   !cfa 12332   ~~> cli 13286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-fz 11682  df-seq 12087  df-exp 12146  df-fac 12333  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290

This theorem is referenced by:  ef0  13704