Theorem ef0lem 12410

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 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))

Assertion

Ref	Expression
ef0lem	⊢ (𝐴 = 0 → seq0( + , 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 ∈ (ℤ≥‘0)) → 𝑘 ∈ (ℤ≥‘0))
2		nn0uz 9940	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrrdi 2332	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → 𝑘 ∈ ℕ0)
4		elnn0 9548	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
5	3, 4	sylib 122	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝑘 ∈ ℕ ∨ 𝑘 = 0))
6		0cnd 8313	. . . . . . . . 9 ⊢ (𝐴 = 0 → 0 ∈ ℂ)
7		eleq1 2301	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ ℂ ↔ 0 ∈ ℂ))
8	6, 7	mpbird 167	. . . . . . . 8 ⊢ (𝐴 = 0 → 𝐴 ∈ ℂ)
9		nnnn0 9553	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
10	9	adantl 277	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
11		efcllem.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))
12	11	eftvalcn 12407	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘)))
13	8, 10, 12	syl2an2r 603	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘)))
14		oveq1 6086	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴↑𝑘) = (0↑𝑘))
15		0exp 10994	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (0↑𝑘) = 0)
16	14, 15	sylan9eq 2291	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) = 0)
17	16	oveq1d 6094	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → ((𝐴↑𝑘) / (!‘𝑘)) = (0 / (!‘𝑘)))
18		faccl 11156	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ)
19		nncn 9295	. . . . . . . . 9 ⊢ ((!‘𝑘) ∈ ℕ → (!‘𝑘) ∈ ℂ)
20		nnap0 9316	. . . . . . . . 9 ⊢ ((!‘𝑘) ∈ ℕ → (!‘𝑘) # 0)
21	19, 20	div0apd 9111	. . . . . . . 8 ⊢ ((!‘𝑘) ∈ ℕ → (0 / (!‘𝑘)) = 0)
22	10, 18, 21	3syl 17	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (0 / (!‘𝑘)) = 0)
23	13, 17, 22	3eqtrd 2275	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 0)
24		nnne0 9315	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
25		velsn 3725	. . . . . . . . . 10 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
26	25	necon3bbii 2457	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ {0} ↔ 𝑘 ≠ 0)
27	24, 26	sylibr 134	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ¬ 𝑘 ∈ {0})
28	27	adantl 277	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → ¬ 𝑘 ∈ {0})
29	28	iffalsed 3650	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → if(𝑘 ∈ {0}, 1, 0) = 0)
30	23, 29	eqtr4d 2274	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = if(𝑘 ∈ {0}, 1, 0))
31		fveq2 5693	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0))
32		0nn0 9561	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
33	11	eftvalcn 12407	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ0) → (𝐹‘0) = ((𝐴↑0) / (!‘0)))
34	8, 32, 33	sylancl 417	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐹‘0) = ((𝐴↑0) / (!‘0)))
35		oveq1 6086	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴↑0) = (0↑0))
36		0exp0e1 10964	. . . . . . . . . . 11 ⊢ (0↑0) = 1
37	35, 36	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (𝐴↑0) = 1)
38	37	oveq1d 6094	. . . . . . . . 9 ⊢ (𝐴 = 0 → ((𝐴↑0) / (!‘0)) = (1 / (!‘0)))
39	34, 38	eqtrd 2271	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐹‘0) = (1 / (!‘0)))
40		fac0 11149	. . . . . . . . . 10 ⊢ (!‘0) = 1
41	40	oveq2i 6090	. . . . . . . . 9 ⊢ (1 / (!‘0)) = (1 / 1)
42		1div1e1 9028	. . . . . . . . 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 ∈ {0})
48	47	iftrued 3647	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝑘 = 0) → if(𝑘 ∈ {0}, 1, 0) = 1)
49	45, 48	eqtr4d 2274	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 = 0) → (𝐹‘𝑘) = if(𝑘 ∈ {0}, 1, 0))
50	30, 49	jaodan 809	. . . 4 ⊢ ((𝐴 = 0 ∧ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) → (𝐹‘𝑘) = if(𝑘 ∈ {0}, 1, 0))
51	5, 50	syldan 282	. . 3 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐹‘𝑘) = if(𝑘 ∈ {0}, 1, 0))
52	32, 2	eleqtri 2313	. . . 4 ⊢ 0 ∈ (ℤ≥‘0)
53	52	a1i 9	. . 3 ⊢ (𝐴 = 0 → 0 ∈ (ℤ≥‘0))
54		1cnd 8336	. . 3 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ {0}) → 1 ∈ ℂ)
55	25	biimpri 133	. . . . . . 7 ⊢ (𝑘 = 0 → 𝑘 ∈ {0})
56	27, 55	orim12i 771	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (¬ 𝑘 ∈ {0} ∨ 𝑘 ∈ {0}))
57	5, 56	syl 14	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (¬ 𝑘 ∈ {0} ∨ 𝑘 ∈ {0}))
58	57	orcomd 741	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝑘 ∈ {0} ∨ ¬ 𝑘 ∈ {0}))
59		df-dc 847	. . . 4 ⊢ (DECID 𝑘 ∈ {0} ↔ (𝑘 ∈ {0} ∨ ¬ 𝑘 ∈ {0}))
60	58, 59	sylibr 134	. . 3 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → DECID 𝑘 ∈ {0})
61		0z 9638	. . . . . 6 ⊢ 0 ∈ ℤ
62		fzsn 10455	. . . . . 6 ⊢ (0 ∈ ℤ → (0...0) = {0})
63	61, 62	ax-mp 5	. . . . 5 ⊢ (0...0) = {0}
64	63	eqimss2i 3305	. . . 4 ⊢ {0} ⊆ (0...0)
65	64	a1i 9	. . 3 ⊢ (𝐴 = 0 → {0} ⊆ (0...0))
66	51, 53, 54, 60, 65	fsum3cvg2 12144	. 2 ⊢ (𝐴 = 0 → seq0( + , 𝐹) ⇝ (seq0( + , 𝐹)‘0))
67	61	a1i 9	. . . 4 ⊢ (𝐴 = 0 → 0 ∈ ℤ)
68	8, 3, 12	syl2an2r 603	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘)))
69		eftcl 12404	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ)
70	8, 3, 69	syl2an2r 603	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ)
71	68, 70	eqeltrd 2315	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐹‘𝑘) ∈ ℂ)
72		addcl 8298	. . . . 5 ⊢ ((𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑘 + 𝑦) ∈ ℂ)
73	72	adantl 277	. . . 4 ⊢ ((𝐴 = 0 ∧ (𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑘 + 𝑦) ∈ ℂ)
74	67, 71, 73	seq3-1 10882	. . 3 ⊢ (𝐴 = 0 → (seq0( + , 𝐹)‘0) = (𝐹‘0))
75	74, 44	eqtrd 2271	. 2 ⊢ (𝐴 = 0 → (seq0( + , 𝐹)‘0) = 1)
76	66, 75	breqtrd 4154	1 ⊢ (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   ⊆ wss 3220  ifcif 3638  {csn 3708   class class class wbr 4128   ↦ cmpt 4190  ‘cfv 5375  (class class class)co 6079  ℂcc 8171  0cc0 8173  1c1 8174   + caddc 8176   / cdiv 8996  ℕcn 9287  ℕ₀cn0 9546  ℤcz 9627  ℤ_≥cuz 9904  ...cfz 10394  seqcseq 10867  ↑cexp 10958  !cfa 11146   ⇝ cli 12027

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-rp 10038  df-fz 10395  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-cj 11590  df-rsqrt 11747  df-abs 11748  df-clim 12028

This theorem is referenced by:  ef0  12422