Theorem expcncf 13242

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 5850	. . . 4 ⊢ (𝑤 = 0 → (𝑥↑𝑤) = (𝑥↑0))
2	1	mpteq2dv 4073	. . 3 ⊢ (𝑤 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2235	. 2 ⊢ (𝑤 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)))
4		oveq2 5850	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑥↑𝑤) = (𝑥↑𝑘))
5	4	mpteq2dv 4073	. . 3 ⊢ (𝑤 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2235	. 2 ⊢ (𝑤 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)))
7		oveq2 5850	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑥↑𝑤) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4073	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2235	. 2 ⊢ (𝑤 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
10		oveq2 5850	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑥↑𝑤) = (𝑥↑𝑁))
11	10	mpteq2dv 4073	. . 3 ⊢ (𝑤 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2235	. 2 ⊢ (𝑤 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)))
13		exp0 10459	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4068	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		ax-1cn 7846	. . . 4 ⊢ 1 ∈ ℂ
16		ssid 3162	. . . 4 ⊢ ℂ ⊆ ℂ
17		cncfmptc 13232	. . . 4 ⊢ ((1 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
18	15, 16, 16, 17	mp3an 1327	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ)
19	14, 18	eqeltri 2239	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)
20		oveq1 5849	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎↑𝑘) = (𝑥↑𝑘))
21	20	cbvmptv 4078	. . . . . 6 ⊢ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))
22	21	eleq1i 2232	. . . . 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 13233	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
26	16, 16, 25	mp2an 423	. . . . . . 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 13241	. . . . 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 10462	. . . . . . . 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 4072	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
33	32	eleq1d 2235	. . . . 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 9305	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136   ⊆ wss 3116   ↦ cmpt 4043  (class class class)co 5842  ℂcc 7751  0cc0 7753  1c1 7754   + caddc 7756   · cmul 7758  ℕ₀cn0 9114  ↑cexp 10454  –cn→ccncf 13207

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  ax-arch 7872  ax-caucvg 7873

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-isom 5197  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-map 6616  df-sup 6949  df-inf 6950  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-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-cncf 13208

This theorem is referenced by: (None)