Theorem ef0lem 13486

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 11010	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleqr 2553	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 10696	. . . . 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 10701	. . . . . . . . 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 13484	. . . . . . . 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 6210	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
12		0exp 12020	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
13	11, 12	sylan9eq 2515	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
14	13	oveq1d 6218	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
15		faccl 12182	. . . . . . . . 9 |- ( k e. NN0 -> ( ! ` k ) e. NN )
16	7, 15	syl 16	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( ! ` k ) e. NN )
17		nncn 10445	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
18		nnne0 10469	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
19	17, 18	div0d 10221	. . . . . . . 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 2499	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
22		nnne0 10469	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
23		elsn 4002	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
24	23	necon3bbii 2713	. . . . . . . . 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 3910	. . . . . . 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 2498	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
30		fveq2 5802	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
31		oveq1 6210	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
32		0exp0e1 11991	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
33	31, 32	syl6eq 2511	. . . . . . . . 9 |- ( A = 0 -> ( A ^ 0 ) = 1 )
34	33	oveq1d 6218	. . . . . . . 8 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
35		0nn0 10709	. . . . . . . . 9 |- 0 e. NN0
36	8	eftval 13484	. . . . . . . . 9 |- ( 0 e. NN0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
37	35, 36	ax-mp 5	. . . . . . . 8 |- ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) )
38		fac0 12175	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
39	38	oveq2i 6214	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
40		1div1e1 10139	. . . . . . . . 9 |- ( 1 / 1 ) = 1
41	39, 40	eqtr2i 2484	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
42	34, 37, 41	3eqtr4g 2520	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
43	30, 42	sylan9eqr 2517	. . . . . 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 3908	. . . . . . 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 2498	. . . . 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 2540	. . . 4 |- 0 e. ( ZZ>= ` 0 )
52	51	a1i 11	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
53		1cnd 9517	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
54		0z 10772	. . . . . 6 |- 0 e. ZZ
55		fzsn 11621	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
56	54, 55	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
57	56	eqimss2i 3522	. . . 4 |- { 0 } C_ ( 0 ... 0 )
58	57	a1i 11	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
59	50, 52, 53, 58	fsumcvg2 13326	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
60	54, 42	seq1i 11941	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
61	59, 60	breqtrd 4427	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   C_ wss 3439   ifcif 3902   {csn 3988   class class class wbr 4403   |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398   + caddc 9400   / cdiv 10108   NNcn 10437   NN0cn0 10694   ZZcz 10761   ZZ>=cuz 10976   ...cfz 11558   seqcseq 11927   ^cexp 11986   !cfa 12172   ~~> cli 13084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-seq 11928  df-exp 11987  df-fac 12173  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088

This theorem is referenced by:  ef0  13498