Theorem ef0lem 12429

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 9959	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrrdi 2332	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → 𝑘 ∈ ℕ0)
4		elnn0 9567	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
5	3, 4	sylib 122	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝑘 ∈ ℕ ∨ 𝑘 = 0))
6		0cnd 8319	. . . . . . . . 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 9572	. . . . . . . . 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 12426	. . . . . . . 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 6092	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴↑𝑘) = (0↑𝑘))
15		0exp 11013	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (0↑𝑘) = 0)
16	14, 15	sylan9eq 2291	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) = 0)
17	16	oveq1d 6100	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝑘 ∈ ℕ) → ((𝐴↑𝑘) / (!‘𝑘)) = (0 / (!‘𝑘)))
18		faccl 11175	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ)
19		nncn 9313	. . . . . . . . 9 ⊢ ((!‘𝑘) ∈ ℕ → (!‘𝑘) ∈ ℂ)
20		nnap0 9334	. . . . . . . . 9 ⊢ ((!‘𝑘) ∈ ℕ → (!‘𝑘) # 0)
21	19, 20	div0apd 9118	. . . . . . . 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 9333	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
25		velsn 3726	. . . . . . . . . 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 5695	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0))
32		0nn0 9580	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
33	11	eftvalcn 12426	. . . . . . . . . 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 6092	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴↑0) = (0↑0))
36		0exp0e1 10983	. . . . . . . . . . 11 ⊢ (0↑0) = 1
37	35, 36	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (𝐴↑0) = 1)
38	37	oveq1d 6100	. . . . . . . . 9 ⊢ (𝐴 = 0 → ((𝐴↑0) / (!‘0)) = (1 / (!‘0)))
39	34, 38	eqtrd 2271	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐹‘0) = (1 / (!‘0)))
40		fac0 11168	. . . . . . . . . 10 ⊢ (!‘0) = 1
41	40	oveq2i 6096	. . . . . . . . 9 ⊢ (1 / (!‘0)) = (1 / 1)
42		1div1e1 9035	. . . . . . . . 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 8342	. . 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 9657	. . . . . 6 ⊢ 0 ∈ ℤ
62		fzsn 10474	. . . . . 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 12163	. 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 12423	. . . . . 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 8304	. . . . 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 10901	. . 3 ⊢ (𝐴 = 0 → (seq0( + , 𝐹)‘0) = (𝐹‘0))
75	74, 44	eqtrd 2271	. 2 ⊢ (𝐴 = 0 → (seq0( + , 𝐹)‘0) = 1)
76	66, 75	breqtrd 4156	1 ⊢ (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   ⊆ wss 3220  ifcif 3638  {csn 3709   class class class wbr 4130   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  ℂcc 8177  0cc0 8179  1c1 8180   + caddc 8182   / cdiv 9003  ℕcn 9305  ℕ₀cn0 9565  ℤcz 9646  ℤ_≥cuz 9923  ...cfz 10413  seqcseq 10886  ↑cexp 10977  !cfa 11165   ⇝ cli 12046

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-rp 10057  df-fz 10414  df-seqfrec 10887  df-exp 10978  df-fac 11166  df-cj 11609  df-rsqrt 11766  df-abs 11767  df-clim 12047

This theorem is used by:  ef0  12441