Theorem expcncf 15523

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 6060	. . . 4 ⊢ (𝑤 = 0 → (𝑥↑𝑤) = (𝑥↑0))
2	1	mpteq2dv 4203	. . 3 ⊢ (𝑤 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2303	. 2 ⊢ (𝑤 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)))
4		oveq2 6060	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑥↑𝑤) = (𝑥↑𝑘))
5	4	mpteq2dv 4203	. . 3 ⊢ (𝑤 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2303	. 2 ⊢ (𝑤 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)))
7		oveq2 6060	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑥↑𝑤) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4203	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2303	. 2 ⊢ (𝑤 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
10		oveq2 6060	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑥↑𝑤) = (𝑥↑𝑁))
11	10	mpteq2dv 4203	. . 3 ⊢ (𝑤 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2303	. 2 ⊢ (𝑤 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)))
13		exp0 10912	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4198	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		ax-1cn 8225	. . . 4 ⊢ 1 ∈ ℂ
16		ssid 3260	. . . 4 ⊢ ℂ ⊆ ℂ
17		cncfmptc 15510	. . . 4 ⊢ ((1 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
18	15, 16, 16, 17	mp3an 1374	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ)
19	14, 18	eqeltri 2307	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)
20		oveq1 6059	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎↑𝑘) = (𝑥↑𝑘))
21	20	cbvmptv 4208	. . . . . 6 ⊢ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))
22	21	eleq1i 2300	. . . . 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 15511	. . . . . . . 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 15522	. . . . 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 10915	. . . . . . . 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 4202	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
33	32	eleq1d 2303	. . . . 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 9698	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ⊆ wss 3213   ↦ cmpt 4173  (class class class)co 6052  ℂcc 8130  0cc0 8132  1c1 8133   + caddc 8135   · cmul 8137  ℕ₀cn0 9501  ↑cexp 10907  –cn→ccncf 15484

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-map 6886  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-rp 9993  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-cncf 15485

This theorem is referenced by: (None)