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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 11234 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | elpw2 5340 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
3 | oveq2 7439 | . . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆)) | |
4 | oveq1 7438 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D𝑛 𝑓) = (𝑆 D𝑛 𝑓)) | |
5 | 4 | fveq1d 6909 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑛)) |
6 | 5 | eleq1d 2824 | . . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ))) |
7 | 3, 6 | rabeqbidv 3452 | . . . 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 5250 | . . 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 25919 | . . 3 ⊢ 𝓑C𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
10 | nn0ex 12530 | . . . 4 ⊢ ℕ0 ∈ V | |
11 | 10 | mptex 7243 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V |
12 | 8, 9, 11 | fvmpt 7016 | . 2 ⊢ (𝑆 ∈ 𝒫 ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) |
13 | 2, 12 | sylbir 235 | 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 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ↑pm cpm 8866 ℂcc 11151 ℕ0cn0 12524 –cn→ccncf 24916 D𝑛 cdvn 25914 𝓑C𝑛ccpn 25915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525 df-cpn 25919 |
This theorem is referenced by: fncpn 25984 elcpn 25985 |
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