Theorem expcncf 13386

Description: The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Assertion

Ref	Expression
expcncf	⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Distinct variable group:   𝑥,𝑁

Proof of Theorem expcncf

Dummy variables 𝑤 𝑘 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5861	. . . 4 ⊢ (𝑤 = 0 → (𝑥↑𝑤) = (𝑥↑0))
2	1	mpteq2dv 4080	. . 3 ⊢ (𝑤 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2239	. 2 ⊢ (𝑤 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)))
4		oveq2 5861	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑥↑𝑤) = (𝑥↑𝑘))
5	4	mpteq2dv 4080	. . 3 ⊢ (𝑤 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2239	. 2 ⊢ (𝑤 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)))
7		oveq2 5861	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑥↑𝑤) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4080	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2239	. 2 ⊢ (𝑤 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
10		oveq2 5861	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑥↑𝑤) = (𝑥↑𝑁))
11	10	mpteq2dv 4080	. . 3 ⊢ (𝑤 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2239	. 2 ⊢ (𝑤 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)))
13		exp0 10480	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4075	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		ax-1cn 7867	. . . 4 ⊢ 1 ∈ ℂ
16		ssid 3167	. . . 4 ⊢ ℂ ⊆ ℂ
17		cncfmptc 13376	. . . 4 ⊢ ((1 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
18	15, 16, 16, 17	mp3an 1332	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ)
19	14, 18	eqeltri 2243	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)
20		oveq1 5860	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎↑𝑘) = (𝑥↑𝑘))
21	20	cbvmptv 4085	. . . . . 6 ⊢ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))
22	21	eleq1i 2236	. . . . 5 ⊢ ((𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
23	22	biimpi 119	. . . . . . 7 ⊢ ((𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
24	23	adantl 275	. . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
25		cncfmptid 13377	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
26	16, 16, 25	mp2an 424	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ)
27	26	a1i 9	. . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
28	24, 27	mulcncf 13385	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ))
29	22, 28	sylan2br 286	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ))
30		expp1 10483	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
31	30	ancoms 266	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
32	31	mpteq2dva 4079	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
33	32	eleq1d 2239	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ)))
34	33	adantr 274	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)) → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ)))
35	29, 34	mpbird 166	. . 3 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ))
36	35	ex 114	. 2 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
37	3, 6, 9, 12, 19, 36	nn0ind 9326	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121   ↦ cmpt 4050  (class class class)co 5853  ℂcc 7772  0cc0 7774  1c1 7775   + caddc 7777   · cmul 7779  ℕ₀cn0 9135  ↑cexp 10475  –cn→ccncf 13351

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-cncf 13352

This theorem is referenced by: (None)