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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 → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10606 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | elpw2 5239 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
3 | oveq2 7153 | . . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆)) | |
4 | oveq1 7152 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D𝑛 𝑓) = (𝑆 D𝑛 𝑓)) | |
5 | 4 | fveq1d 6665 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑛)) |
6 | 5 | eleq1d 2894 | . . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ))) |
7 | 3, 6 | rabeqbidv 3483 | . . . 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 5153 | . . 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 24394 | . . 3 ⊢ 𝓑C𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
10 | nn0ex 11891 | . . . 4 ⊢ ℕ0 ∈ V | |
11 | 10 | mptex 6977 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V |
12 | 8, 9, 11 | fvmpt 6761 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
13 | 2, 12 | sylbir 236 | 1 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘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 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 𝒫 cpw 4535 ↦ cmpt 5137 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ↑pm cpm 8396 ℂcc 10523 ℕ0cn0 11885 –cn→ccncf 23411 D𝑛 cdvn 24389 𝓑C𝑛ccpn 24390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-n0 11886 df-cpn 24394 |
This theorem is referenced by: fncpn 24457 elcpn 24458 |
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