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Theorem dvnf 25948

Description: The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
dvnf	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)

**Proof of Theorem dvnf**
Step	Hyp	Ref	Expression
1		dvnff 25944	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
2	1	ffvelcdmda 7098	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
3	2	3impa 1107	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
4		cnex 11239	. . . 4 ⊢ ℂ ∈ V
5		simp2 1134	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
6	5	dmexd 7916	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom 𝐹 ∈ V)
7		elpm2g 8873	. . . 4 ⊢ ((ℂ ∈ V ∧ dom 𝐹 ∈ V) → (((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹) ↔ (((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ ∧ dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)))
8	4, 6, 7	sylancr 585	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹) ↔ (((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ ∧ dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)))
9	3, 8	mpbid 231	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ ∧ dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹))
10	9	simpld 493	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 {cpr 4635 dom cdm 5682 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ↑_pm cpm 8856 ℂcc 11156 ℝcr 11157 ℕ₀cn0 12524 D^𝑛 cdvn 25884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fi 9454 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-icc 13385 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-rest 17437 df-topn 17438 df-topgen 17458 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cnp 23223 df-haus 23310 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-limc 25886 df-dv 25887 df-dvn 25888

This theorem is referenced by: dvn2bss 25951 dvnres 25952 cpnord 25956 taylfvallem1 26384 tayl0 26389 taylply2 26395 taylply2OLD 26396 taylply 26397 dvtaylp 26398 dvntaylp 26399 dvntaylp0 26400 taylthlem1 26401 taylthlem2 26402 taylthlem2OLD 26403

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