Theorem ef0lem 11366

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 109	. . . . . 6 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. ( ZZ>= ` 0 ) )
2		nn0uz 9360	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtrrdi 2233	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 8979	. . . . 5 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
5	3, 4	sylib 121	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. NN \/ k = 0 ) )
6		0cnd 7759	. . . . . . . . 9 |- ( A = 0 -> 0 e. CC )
7		eleq1 2202	. . . . . . . . 9 |- ( A = 0 -> ( A e. CC <-> 0 e. CC ) )
8	6, 7	mpbird 166	. . . . . . . 8 |- ( A = 0 -> A e. CC )
9		nnnn0 8984	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
10	9	adantl 275	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> k e. NN0 )
11		efcllem.1	. . . . . . . . 9 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
12	11	eftvalcn 11363	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13	8, 10, 12	syl2an2r 584	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
14		oveq1 5781	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
15		0exp 10328	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
16	14, 15	sylan9eq 2192	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
17	16	oveq1d 5789	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
18		faccl 10481	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
19		nncn 8728	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
20		nnap0 8749	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) # 0 )
21	19, 20	div0apd 8547	. . . . . . . 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 2176	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
24		nnne0 8748	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
25		velsn 3544	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
26	25	necon3bbii 2345	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
27	24, 26	sylibr 133	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
28	27	adantl 275	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
29	28	iffalsed 3484	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
30	23, 29	eqtr4d 2175	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
31		fveq2 5421	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
32		0nn0 8992	. . . . . . . . . 10 |- 0 e. NN0
33	11	eftvalcn 11363	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
34	8, 32, 33	sylancl 409	. . . . . . . . 9 |- ( A = 0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35		oveq1 5781	. . . . . . . . . . 11 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
36		0exp0e1 10298	. . . . . . . . . . 11 |- ( 0 ^ 0 ) = 1
37	35, 36	syl6eq 2188	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = 1 )
38	37	oveq1d 5789	. . . . . . . . 9 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
39	34, 38	eqtrd 2172	. . . . . . . 8 |- ( A = 0 -> ( F ` 0 ) = ( 1 / ( ! ` 0 ) ) )
40		fac0 10474	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
41	40	oveq2i 5785	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
42		1div1e1 8464	. . . . . . . . 9 |- ( 1 / 1 ) = 1
43	41, 42	eqtr2i 2161	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
44	39, 43	syl6eqr 2190	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
45	31, 44	sylan9eqr 2194	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
46		simpr 109	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
47	46, 25	sylibr 133	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
48	47	iftrued 3481	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
49	45, 48	eqtr4d 2175	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	30, 49	jaodan 786	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
51	5, 50	syldan 280	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
52	32, 2	eleqtri 2214	. . . 4 |- 0 e. ( ZZ>= ` 0 )
53	52	a1i 9	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
54		1cnd 7782	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
55	25	biimpri 132	. . . . . . 7 |- ( k = 0 -> k e. { 0 } )
56	27, 55	orim12i 748	. . . . . 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 718	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. { 0 } \/ -. k e. { 0 } ) )
59		df-dc 820	. . . 4 |- (DECID k e. { 0 } <-> ( k e. { 0 } \/ -. k e. { 0 } ) )
60	58, 59	sylibr 133	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> DECID k e. { 0 } )
61		0z 9065	. . . . . 6 |- 0 e. ZZ
62		fzsn 9846	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
63	61, 62	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
64	63	eqimss2i 3154	. . . 4 |- { 0 } C_ ( 0 ... 0 )
65	64	a1i 9	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
66	51, 53, 54, 60, 65	fsum3cvg2 11163	. 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 584	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
69		eftcl 11360	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
70	8, 3, 69	syl2an2r 584	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
71	68, 70	eqeltrd 2216	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
72		addcl 7745	. . . . 5 |- ( ( k e. CC /\ y e. CC ) -> ( k + y ) e. CC )
73	72	adantl 275	. . . 4 |- ( ( A = 0 /\ ( k e. CC /\ y e. CC ) ) -> ( k + y ) e. CC )
74	67, 71, 73	seq3-1 10233	. . 3 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = ( F ` 0 ) )
75	74, 44	eqtrd 2172	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
76	66, 75	breqtrd 3954	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308   C_ wss 3071   ifcif 3474   {csn 3527   class class class wbr 3929   |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790   seqcseq 10218   ^cexp 10292   !cfa 10471   ~~> cli 11047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-fz 9791  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-cj 10614  df-rsqrt 10770  df-abs 10771  df-clim 11048

This theorem is referenced by:  ef0  11378