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Mirrors > Home > MPE Home > Th. List > cpnfval | Structured version Visualization version GIF version |
Description: Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
cpnfval | ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10055 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | elpw2 4858 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
3 | oveq2 6698 | . . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆)) | |
4 | oveq1 6697 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D𝑛 𝑓) = (𝑆 D𝑛 𝑓)) | |
5 | 4 | fveq1d 6231 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑛)) |
6 | 5 | eleq1d 2715 | . . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ))) |
7 | 3, 6 | rabeqbidv 3226 | . . . 4 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) |
8 | 7 | mpteq2dv 4778 | . . 3 ⊢ (𝑠 = 𝑆 → (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
9 | df-cpn 23678 | . . 3 ⊢ Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
10 | nn0ex 11336 | . . . 4 ⊢ ℕ0 ∈ V | |
11 | 10 | mptex 6527 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V |
12 | 8, 9, 11 | fvmpt 6321 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
13 | 2, 12 | sylbir 225 | 1 ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 {crab 2945 ⊆ wss 3607 𝒫 cpw 4191 ↦ cmpt 4762 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 ↑pm cpm 7900 ℂcc 9972 ℕ0cn0 11330 –cn→ccncf 22726 D𝑛 cdvn 23673 Cnccpn 23674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-n0 11331 df-cpn 23678 |
This theorem is referenced by: fncpn 23741 elcpn 23742 |
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