Theorem ef0lem 12427

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
efcllem.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 variables k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. ( ZZ>= ` 0 ) )
2		nn0uz 9957	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtrrdi 2332	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 9565	. . . . 5 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
5	3, 4	sylib 122	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. NN \/ k = 0 ) )
6		0cnd 8319	. . . . . . . . 9 |- ( A = 0 -> 0 e. CC )
7		eleq1 2301	. . . . . . . . 9 |- ( A = 0 -> ( A e. CC <-> 0 e. CC ) )
8	6, 7	mpbird 167	. . . . . . . 8 |- ( A = 0 -> A e. CC )
9		nnnn0 9570	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
10	9	adantl 277	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> k e. NN0 )
11		efcllem.1	. . . . . . . . 9 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
12	11	eftvalcn 12424	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13	8, 10, 12	syl2an2r 603	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
14		oveq1 6092	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
15		0exp 11011	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
16	14, 15	sylan9eq 2291	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
17	16	oveq1d 6100	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
18		faccl 11173	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
19		nncn 9312	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
20		nnap0 9333	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) # 0 )
21	19, 20	div0apd 9117	. . . . . . . 8 |- ( ( ! ` k ) e. NN -> ( 0 / ( ! ` k ) ) = 0 )
22	10, 18, 21	3syl 17	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( 0 / ( ! ` k ) ) = 0 )
23	13, 17, 22	3eqtrd 2275	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
24		nnne0 9332	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
25		velsn 3726	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
26	25	necon3bbii 2457	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
27	24, 26	sylibr 134	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
28	27	adantl 277	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
29	28	iffalsed 3650	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
30	23, 29	eqtr4d 2274	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
31		fveq2 5695	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
32		0nn0 9578	. . . . . . . . . 10 |- 0 e. NN0
33	11	eftvalcn 12424	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
34	8, 32, 33	sylancl 417	. . . . . . . . 9 |- ( A = 0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35		oveq1 6092	. . . . . . . . . . 11 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
36		0exp0e1 10981	. . . . . . . . . . 11 |- ( 0 ^ 0 ) = 1
37	35, 36	eqtrdi 2287	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = 1 )
38	37	oveq1d 6100	. . . . . . . . 9 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
39	34, 38	eqtrd 2271	. . . . . . . 8 |- ( A = 0 -> ( F ` 0 ) = ( 1 / ( ! ` 0 ) ) )
40		fac0 11166	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
41	40	oveq2i 6096	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
42		1div1e1 9034	. . . . . . . . 9 |- ( 1 / 1 ) = 1
43	41, 42	eqtr2i 2260	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
44	39, 43	eqtr4di 2289	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
45	31, 44	sylan9eqr 2293	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
46		simpr 110	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
47	46, 25	sylibr 134	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
48	47	iftrued 3647	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
49	45, 48	eqtr4d 2274	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	30, 49	jaodan 809	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
51	5, 50	syldan 282	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
52	32, 2	eleqtri 2313	. . . 4 |- 0 e. ( ZZ>= ` 0 )
53	52	a1i 9	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
54		1cnd 8342	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
55	25	biimpri 133	. . . . . . 7 |- ( k = 0 -> k e. { 0 } )
56	27, 55	orim12i 771	. . . . . 6 |- ( ( k e. NN \/ k = 0 ) -> ( -. k e. { 0 } \/ k e. { 0 } ) )
57	5, 56	syl 14	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( -. k e. { 0 } \/ k e. { 0 } ) )
58	57	orcomd 741	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. { 0 } \/ -. k e. { 0 } ) )
59		df-dc 847	. . . 4 |- (DECID k e. { 0 } <-> ( k e. { 0 } \/ -. k e. { 0 } ) )
60	58, 59	sylibr 134	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> DECID k e. { 0 } )
61		0z 9655	. . . . . 6 |- 0 e. ZZ
62		fzsn 10472	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
63	61, 62	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
64	63	eqimss2i 3305	. . . 4 |- { 0 } C_ ( 0 ... 0 )
65	64	a1i 9	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
66	51, 53, 54, 60, 65	fsum3cvg2 12161	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
67	61	a1i 9	. . . 4 |- ( A = 0 -> 0 e. ZZ )
68	8, 3, 12	syl2an2r 603	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
69		eftcl 12421	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
70	8, 3, 69	syl2an2r 603	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
71	68, 70	eqeltrd 2315	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
72		addcl 8304	. . . . 5 |- ( ( k e. CC /\ y e. CC ) -> ( k + y ) e. CC )
73	72	adantl 277	. . . 4 |- ( ( A = 0 /\ ( k e. CC /\ y e. CC ) ) -> ( k + y ) e. CC )
74	67, 71, 73	seq3-1 10899	. . 3 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = ( F ` 0 ) )
75	74, 44	eqtrd 2271	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
76	66, 75	breqtrd 4156	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   C_ wss 3220   ifcif 3638   {csn 3709   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884   ^cexp 10975   !cfa 11163   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-fz 10412  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-cj 11607  df-rsqrt 11764  df-abs 11765  df-clim 12045

This theorem is used by:  ef0  12439