Theorem ef0lem 11825

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 9636	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtrrdi 2290	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 9251	. . . . 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 8019	. . . . . . . . 9 |- ( A = 0 -> 0 e. CC )
7		eleq1 2259	. . . . . . . . 9 |- ( A = 0 -> ( A e. CC <-> 0 e. CC ) )
8	6, 7	mpbird 167	. . . . . . . 8 |- ( A = 0 -> A e. CC )
9		nnnn0 9256	. . . . . . . . 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 11822	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13	8, 10, 12	syl2an2r 595	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
14		oveq1 5929	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
15		0exp 10666	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
16	14, 15	sylan9eq 2249	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
17	16	oveq1d 5937	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
18		faccl 10827	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
19		nncn 8998	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
20		nnap0 9019	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) # 0 )
21	19, 20	div0apd 8814	. . . . . . . 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 2233	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
24		nnne0 9018	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
25		velsn 3639	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
26	25	necon3bbii 2404	. . . . . . . . 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 3571	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
30	23, 29	eqtr4d 2232	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
31		fveq2 5558	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
32		0nn0 9264	. . . . . . . . . 10 |- 0 e. NN0
33	11	eftvalcn 11822	. . . . . . . . . 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 5929	. . . . . . . . . . 11 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
36		0exp0e1 10636	. . . . . . . . . . 11 |- ( 0 ^ 0 ) = 1
37	35, 36	eqtrdi 2245	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = 1 )
38	37	oveq1d 5937	. . . . . . . . 9 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
39	34, 38	eqtrd 2229	. . . . . . . 8 |- ( A = 0 -> ( F ` 0 ) = ( 1 / ( ! ` 0 ) ) )
40		fac0 10820	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
41	40	oveq2i 5933	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
42		1div1e1 8731	. . . . . . . . 9 |- ( 1 / 1 ) = 1
43	41, 42	eqtr2i 2218	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
44	39, 43	eqtr4di 2247	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
45	31, 44	sylan9eqr 2251	. . . . . 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 3568	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
49	45, 48	eqtr4d 2232	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	30, 49	jaodan 798	. . . 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 2271	. . . 4 |- 0 e. ( ZZ>= ` 0 )
53	52	a1i 9	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
54		1cnd 8042	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
55	25	biimpri 133	. . . . . . 7 |- ( k = 0 -> k e. { 0 } )
56	27, 55	orim12i 760	. . . . . 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 730	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. { 0 } \/ -. k e. { 0 } ) )
59		df-dc 836	. . . 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 9337	. . . . . 6 |- 0 e. ZZ
62		fzsn 10141	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
63	61, 62	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
64	63	eqimss2i 3240	. . . 4 |- { 0 } C_ ( 0 ... 0 )
65	64	a1i 9	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
66	51, 53, 54, 60, 65	fsum3cvg2 11559	. 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 595	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
69		eftcl 11819	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
70	8, 3, 69	syl2an2r 595	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
71	68, 70	eqeltrd 2273	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
72		addcl 8004	. . . . 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 10554	. . 3 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = ( F ` 0 ) )
75	74, 44	eqtrd 2229	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
76	66, 75	breqtrd 4059	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   C_ wss 3157   ifcif 3561   {csn 3622   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539   ^cexp 10630   !cfa 10817   ~~> cli 11443

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-fz 10084  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-cj 11007  df-rsqrt 11163  df-abs 11164  df-clim 11444

This theorem is referenced by:  ef0  11837