Theorem ef0lem 12350

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 9892	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtrrdi 2328	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 9500	. . . . 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 8269	. . . . . . . . 9 |- ( A = 0 -> 0 e. CC )
7		eleq1 2297	. . . . . . . . 9 |- ( A = 0 -> ( A e. CC <-> 0 e. CC ) )
8	6, 7	mpbird 167	. . . . . . . 8 |- ( A = 0 -> A e. CC )
9		nnnn0 9505	. . . . . . . . 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 12347	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13	8, 10, 12	syl2an2r 599	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
14		oveq1 6059	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
15		0exp 10940	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
16	14, 15	sylan9eq 2287	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
17	16	oveq1d 6067	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
18		faccl 11101	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
19		nncn 9247	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
20		nnap0 9268	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) # 0 )
21	19, 20	div0apd 9063	. . . . . . . 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 2271	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
24		nnne0 9267	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
25		velsn 3708	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
26	25	necon3bbii 2451	. . . . . . . . 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 3634	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
30	23, 29	eqtr4d 2270	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
31		fveq2 5672	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
32		0nn0 9513	. . . . . . . . . 10 |- 0 e. NN0
33	11	eftvalcn 12347	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
34	8, 32, 33	sylancl 413	. . . . . . . . 9 |- ( A = 0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35		oveq1 6059	. . . . . . . . . . 11 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
36		0exp0e1 10910	. . . . . . . . . . 11 |- ( 0 ^ 0 ) = 1
37	35, 36	eqtrdi 2283	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = 1 )
38	37	oveq1d 6067	. . . . . . . . 9 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
39	34, 38	eqtrd 2267	. . . . . . . 8 |- ( A = 0 -> ( F ` 0 ) = ( 1 / ( ! ` 0 ) ) )
40		fac0 11094	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
41	40	oveq2i 6063	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
42		1div1e1 8980	. . . . . . . . 9 |- ( 1 / 1 ) = 1
43	41, 42	eqtr2i 2256	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
44	39, 43	eqtr4di 2285	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
45	31, 44	sylan9eqr 2289	. . . . . 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 3631	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
49	45, 48	eqtr4d 2270	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	30, 49	jaodan 805	. . . 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 2309	. . . 4 |- 0 e. ( ZZ>= ` 0 )
53	52	a1i 9	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
54		1cnd 8292	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
55	25	biimpri 133	. . . . . . 7 |- ( k = 0 -> k e. { 0 } )
56	27, 55	orim12i 767	. . . . . 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 737	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. { 0 } \/ -. k e. { 0 } ) )
59		df-dc 843	. . . 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 9590	. . . . . 6 |- 0 e. ZZ
62		fzsn 10403	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
63	61, 62	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
64	63	eqimss2i 3297	. . . 4 |- { 0 } C_ ( 0 ... 0 )
65	64	a1i 9	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
66	51, 53, 54, 60, 65	fsum3cvg2 12084	. 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 599	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
69		eftcl 12344	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
70	8, 3, 69	syl2an2r 599	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
71	68, 70	eqeltrd 2311	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
72		addcl 8254	. . . . 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 10828	. . 3 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = ( F ` 0 ) )
75	74, 44	eqtrd 2267	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
76	66, 75	breqtrd 4137	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   C_ wss 3213   ifcif 3622   {csn 3691   class class class wbr 4111   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   / cdiv 8948   NNcn 9239   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   seqcseq 10813   ^cexp 10904   !cfa 11091   ~~> cli 11967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-n0 9499  df-z 9580  df-uz 9857  df-rp 9990  df-fz 10346  df-seqfrec 10814  df-exp 10905  df-fac 11092  df-cj 11531  df-rsqrt 11687  df-abs 11688  df-clim 11968

This theorem is referenced by:  ef0  12362