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Mirrors > Home > MPE Home > Th. List > elcpn | 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 |
---|---|
elcpn | ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpnfval 25312 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
2 | 1 | fveq1d 6849 | . . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝓑C𝑛‘𝑆)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})‘𝑁)) |
3 | fveq2 6847 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑁)) | |
4 | 3 | eleq1d 2823 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ))) |
5 | 4 | rabbidv 3418 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)}) |
6 | eqid 2737 | . . . . 5 ⊢ (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)}) | |
7 | ovex 7395 | . . . . . 6 ⊢ (ℂ ↑pm 𝑆) ∈ V | |
8 | 7 | rabex 5294 | . . . . 5 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)} ∈ V |
9 | 5, 6, 8 | fvmpt 6953 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)}) |
10 | 2, 9 | sylan9eq 2797 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝓑C𝑛‘𝑆)‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)}) |
11 | 10 | eleq2d 2824 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ 𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)})) |
12 | oveq2 7370 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑆 D𝑛 𝑓) = (𝑆 D𝑛 𝐹)) | |
13 | 12 | fveq1d 6849 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑆 D𝑛 𝑓)‘𝑁) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
14 | dmeq 5864 | . . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
15 | 14 | oveq1d 7377 | . . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓–cn→ℂ) = (dom 𝐹–cn→ℂ)) |
16 | 13, 15 | eleq12d 2832 | . . 3 ⊢ (𝑓 = 𝐹 → (((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))) |
17 | 16 | elrab 3650 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)} ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))) |
18 | 11, 17 | bitrdi 287 | 1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3410 ⊆ wss 3915 ↦ cmpt 5193 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 ↑pm cpm 8773 ℂcc 11056 ℕ0cn0 12420 –cn→ccncf 24255 D𝑛 cdvn 25244 𝓑C𝑛ccpn 25245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-1cn 11116 ax-addcl 11118 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-n0 12421 df-cpn 25249 |
This theorem is referenced by: cpnord 25315 cpncn 25316 cpnres 25317 c1lip2 25378 plycpn 25665 |
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