Theorem expcncf 15336

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 6026	. . . 4 ⊢ (𝑤 = 0 → (𝑥↑𝑤) = (𝑥↑0))
2	1	mpteq2dv 4180	. . 3 ⊢ (𝑤 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2300	. 2 ⊢ (𝑤 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)))
4		oveq2 6026	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑥↑𝑤) = (𝑥↑𝑘))
5	4	mpteq2dv 4180	. . 3 ⊢ (𝑤 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2300	. 2 ⊢ (𝑤 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)))
7		oveq2 6026	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑥↑𝑤) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4180	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2300	. 2 ⊢ (𝑤 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
10		oveq2 6026	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑥↑𝑤) = (𝑥↑𝑁))
11	10	mpteq2dv 4180	. . 3 ⊢ (𝑤 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2300	. 2 ⊢ (𝑤 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)))
13		exp0 10806	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4175	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		ax-1cn 8125	. . . 4 ⊢ 1 ∈ ℂ
16		ssid 3247	. . . 4 ⊢ ℂ ⊆ ℂ
17		cncfmptc 15323	. . . 4 ⊢ ((1 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
18	15, 16, 16, 17	mp3an 1373	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ)
19	14, 18	eqeltri 2304	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)
20		oveq1 6025	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎↑𝑘) = (𝑥↑𝑘))
21	20	cbvmptv 4185	. . . . . 6 ⊢ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))
22	21	eleq1i 2297	. . . . 5 ⊢ ((𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
23	22	biimpi 120	. . . . . . 7 ⊢ ((𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
24	23	adantl 277	. . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
25		cncfmptid 15324	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
26	16, 16, 25	mp2an 426	. . . . . . 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 15335	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ))
29	22, 28	sylan2br 288	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ))
30		expp1 10809	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
31	30	ancoms 268	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑘 + 1)) = ((𝑥↑𝑘) · 𝑥))
32	31	mpteq2dva 4179	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
33	32	eleq1d 2300	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)) ∈ (ℂ–cn→ℂ)))
34	33	adantr 276	. . . 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 167	. . 3 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ))
36	35	ex 115	. 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 9594	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202   ⊆ wss 3200   ↦ cmpt 4150  (class class class)co 6018  ℂcc 8030  0cc0 8032  1c1 8033   + caddc 8035   · cmul 8037  ℕ₀cn0 9402  ↑cexp 10801  –cn→ccncf 15297

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-cncf 15298

This theorem is referenced by: (None)