Theorem ef0lem 14809

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 477	. . . . . 6 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. ( ZZ>= ` 0 ) )
2		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleqr 2712	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 11294	. . . . 5 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
5	3, 4	sylib 208	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. NN \/ k = 0 ) )
6		nnnn0 11299	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
7	6	adantl 482	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> k e. NN0 )
8		eftval.1	. . . . . . . . 9 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
9	8	eftval 14807	. . . . . . . 8 |- ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
10	7, 9	syl 17	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
11		oveq1 6657	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
12		0exp 12895	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
13	11, 12	sylan9eq 2676	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
14	13	oveq1d 6665	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
15		faccl 13070	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
16		nncn 11028	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
17		nnne0 11053	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
18	16, 17	div0d 10800	. . . . . . . 8 |- ( ( ! ` k ) e. NN -> ( 0 / ( ! ` k ) ) = 0 )
19	7, 15, 18	3syl 18	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( 0 / ( ! ` k ) ) = 0 )
20	10, 14, 19	3eqtrd 2660	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
21		nnne0 11053	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
22		velsn 4193	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
23	22	necon3bbii 2841	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
24	21, 23	sylibr 224	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
25	24	adantl 482	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
26	25	iffalsed 4097	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
27	20, 26	eqtr4d 2659	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
28		fveq2 6191	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
29		oveq1 6657	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
30		0exp0e1 12865	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
31	29, 30	syl6eq 2672	. . . . . . . . 9 |- ( A = 0 -> ( A ^ 0 ) = 1 )
32	31	oveq1d 6665	. . . . . . . 8 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
33		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
34	8	eftval 14807	. . . . . . . . 9 |- ( 0 e. NN0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35	33, 34	ax-mp 5	. . . . . . . 8 |- ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) )
36		fac0 13063	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
37	36	oveq2i 6661	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
38		1div1e1 10717	. . . . . . . . 9 |- ( 1 / 1 ) = 1
39	37, 38	eqtr2i 2645	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
40	32, 35, 39	3eqtr4g 2681	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
41	28, 40	sylan9eqr 2678	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
42		simpr 477	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
43	42, 22	sylibr 224	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
44	43	iftrued 4094	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
45	41, 44	eqtr4d 2659	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
46	27, 45	jaodan 826	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
47	5, 46	syldan 487	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
48	33, 2	eleqtri 2699	. . . 4 |- 0 e. ( ZZ>= ` 0 )
49	48	a1i 11	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
50		1cnd 10056	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
51		0z 11388	. . . . . 6 |- 0 e. ZZ
52		fzsn 12383	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
53	51, 52	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
54	53	eqimss2i 3660	. . . 4 |- { 0 } C_ ( 0 ... 0 )
55	54	a1i 11	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
56	47, 49, 50, 55	fsumcvg2 14458	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
57	51, 40	seq1i 12815	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
58	56, 57	breqtrd 4679	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ^cexp 12860   !cfa 13060   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-fac 13061  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  ef0  14821