Theorem ef0lem 11601

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 9500	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtrrdi 2260	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 9116	. . . . 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 7892	. . . . . . . . 9 |- ( A = 0 -> 0 e. CC )
7		eleq1 2229	. . . . . . . . 9 |- ( A = 0 -> ( A e. CC <-> 0 e. CC ) )
8	6, 7	mpbird 166	. . . . . . . 8 |- ( A = 0 -> A e. CC )
9		nnnn0 9121	. . . . . . . . 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 11598	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13	8, 10, 12	syl2an2r 585	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
14		oveq1 5849	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
15		0exp 10490	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
16	14, 15	sylan9eq 2219	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
17	16	oveq1d 5857	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
18		faccl 10648	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
19		nncn 8865	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
20		nnap0 8886	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) # 0 )
21	19, 20	div0apd 8683	. . . . . . . 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 2202	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
24		nnne0 8885	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
25		velsn 3593	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
26	25	necon3bbii 2373	. . . . . . . . 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 3530	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
30	23, 29	eqtr4d 2201	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
31		fveq2 5486	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
32		0nn0 9129	. . . . . . . . . 10 |- 0 e. NN0
33	11	eftvalcn 11598	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
34	8, 32, 33	sylancl 410	. . . . . . . . 9 |- ( A = 0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35		oveq1 5849	. . . . . . . . . . 11 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
36		0exp0e1 10460	. . . . . . . . . . 11 |- ( 0 ^ 0 ) = 1
37	35, 36	eqtrdi 2215	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = 1 )
38	37	oveq1d 5857	. . . . . . . . 9 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
39	34, 38	eqtrd 2198	. . . . . . . 8 |- ( A = 0 -> ( F ` 0 ) = ( 1 / ( ! ` 0 ) ) )
40		fac0 10641	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
41	40	oveq2i 5853	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
42		1div1e1 8600	. . . . . . . . 9 |- ( 1 / 1 ) = 1
43	41, 42	eqtr2i 2187	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
44	39, 43	eqtr4di 2217	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
45	31, 44	sylan9eqr 2221	. . . . . 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 3527	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
49	45, 48	eqtr4d 2201	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
50	30, 49	jaodan 787	. . . 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 2241	. . . 4 |- 0 e. ( ZZ>= ` 0 )
53	52	a1i 9	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
54		1cnd 7915	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
55	25	biimpri 132	. . . . . . 7 |- ( k = 0 -> k e. { 0 } )
56	27, 55	orim12i 749	. . . . . 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 719	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. { 0 } \/ -. k e. { 0 } ) )
59		df-dc 825	. . . 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 9202	. . . . . 6 |- 0 e. ZZ
62		fzsn 10001	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
63	61, 62	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
64	63	eqimss2i 3199	. . . 4 |- { 0 } C_ ( 0 ... 0 )
65	64	a1i 9	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
66	51, 53, 54, 60, 65	fsum3cvg2 11335	. 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 585	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
69		eftcl 11595	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
70	8, 3, 69	syl2an2r 585	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
71	68, 70	eqeltrd 2243	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
72		addcl 7878	. . . . 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 10395	. . 3 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = ( F ` 0 ) )
75	74, 44	eqtrd 2198	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
76	66, 75	breqtrd 4008	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336   C_ wss 3116   ifcif 3520   {csn 3576   class class class wbr 3982   |-> cmpt 4043   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   / cdiv 8568   NNcn 8857   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380   ^cexp 10454   !cfa 10638   ~~> cli 11219

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-fz 9945  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-cj 10784  df-rsqrt 10940  df-abs 10941  df-clim 11220

This theorem is referenced by:  ef0  11613