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Theorem dvnf 25678

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 25674	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
2	1	ffvelcdmda 7087	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
3	2	3impa 1108	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
4		cnex 11195	. . . 4 ⊢ ℂ ∈ V
5		simp2 1135	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
6	5	dmexd 7900	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom 𝐹 ∈ V)
7		elpm2g 8842	. . . 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 1085 ∈ wcel 2104 Vcvv 3472 ⊆ wss 3949 {cpr 4631 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 ↑_pm cpm 8825 ℂcc 11112 ℝcr 11113 ℕ₀cn0 12478 D^𝑛 cdvn 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9410 df-sup 9441 df-inf 9442 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-icc 13337 df-fz 13491 df-seq 13973 df-exp 14034 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-topgen 17395 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-fbas 21143 df-fg 21144 df-cnfld 21147 df-top 22618 df-topon 22635 df-topsp 22657 df-bases 22671 df-cld 22745 df-ntr 22746 df-cls 22747 df-nei 22824 df-lp 22862 df-perf 22863 df-cnp 22954 df-haus 23041 df-fil 23572 df-fm 23664 df-flim 23665 df-flf 23666 df-xms 24048 df-ms 24049 df-limc 25617 df-dv 25618 df-dvn 25619

This theorem is referenced by: dvn2bss 25681 dvnres 25682 cpnord 25686 taylfvallem1 26103 tayl0 26108 taylply2 26114 taylply 26115 dvtaylp 26116 dvntaylp 26117 dvntaylp0 26118 taylthlem1 26119 taylthlem2 26120

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