Theorem expcncf 13759

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 5877	. . . 4 ⊢ (𝑤 = 0 → (𝑥↑𝑤) = (𝑥↑0))
2	1	mpteq2dv 4091	. . 3 ⊢ (𝑤 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2246	. 2 ⊢ (𝑤 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)))
4		oveq2 5877	. . . 4 ⊢ (𝑤 = 𝑘 → (𝑥↑𝑤) = (𝑥↑𝑘))
5	4	mpteq2dv 4091	. . 3 ⊢ (𝑤 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2246	. 2 ⊢ (𝑤 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ)))
7		oveq2 5877	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (𝑥↑𝑤) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4091	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2246	. 2 ⊢ (𝑤 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (ℂ–cn→ℂ)))
10		oveq2 5877	. . . 4 ⊢ (𝑤 = 𝑁 → (𝑥↑𝑤) = (𝑥↑𝑁))
11	10	mpteq2dv 4091	. . 3 ⊢ (𝑤 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2246	. 2 ⊢ (𝑤 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑤)) ∈ (ℂ–cn→ℂ) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)))
13		exp0 10510	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4086	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		ax-1cn 7895	. . . 4 ⊢ 1 ∈ ℂ
16		ssid 3175	. . . 4 ⊢ ℂ ⊆ ℂ
17		cncfmptc 13749	. . . 4 ⊢ ((1 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
18	15, 16, 16, 17	mp3an 1337	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ)
19	14, 18	eqeltri 2250	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (ℂ–cn→ℂ)
20		oveq1 5876	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎↑𝑘) = (𝑥↑𝑘))
21	20	cbvmptv 4096	. . . . . 6 ⊢ (𝑎 ∈ ℂ ↦ (𝑎↑𝑘)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))
22	21	eleq1i 2243	. . . . 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 13750	. . . . . . . 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 13758	. . . . 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 10513	. . . . . . . 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 4090	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑥 ∈ ℂ ↦ ((𝑥↑𝑘) · 𝑥)))
33	32	eleq1d 2246	. . . . 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 9356	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ⊆ wss 3129   ↦ cmpt 4061  (class class class)co 5869  ℂcc 7800  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807  ℕ₀cn0 9165  ↑cexp 10505  –cn→ccncf 13724

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-map 6644  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-cncf 13725

This theorem is referenced by: (None)