dvnf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvnf		Structured version Visualization version GIF version

Theorem dvnf 25189

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 25185	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
2	1	ffvelcdmda 7011	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
3	2	3impa 1109	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm dom 𝐹))
4		cnex 11045	. . . 4 ⊢ ℂ ∈ V
5		simp2 1136	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
6	5	dmexd 7812	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom 𝐹 ∈ V)
7		elpm2g 8695	. . . 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 587	. . 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 495	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 396 ∧ w3a 1086 ∈ wcel 2105 Vcvv 3441 ⊆ wss 3897 {cpr 4574 dom cdm 5614 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 ↑_pm cpm 8679 ℂcc 10962 ℝcr 10963 ℕ₀cn0 12326 D^𝑛 cdvn 25126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-pm 8681 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fi 9260 df-sup 9291 df-inf 9292 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-q 12782 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-icc 13179 df-fz 13333 df-seq 13815 df-exp 13876 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-struct 16937 df-slot 16972 df-ndx 16984 df-base 17002 df-plusg 17064 df-mulr 17065 df-starv 17066 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-rest 17222 df-topn 17223 df-topgen 17243 df-psmet 20687 df-xmet 20688 df-met 20689 df-bl 20690 df-mopn 20691 df-fbas 20692 df-fg 20693 df-cnfld 20696 df-top 22141 df-topon 22158 df-topsp 22180 df-bases 22194 df-cld 22268 df-ntr 22269 df-cls 22270 df-nei 22347 df-lp 22385 df-perf 22386 df-cnp 22477 df-haus 22564 df-fil 23095 df-fm 23187 df-flim 23188 df-flf 23189 df-xms 23571 df-ms 23572 df-limc 25128 df-dv 25129 df-dvn 25130

This theorem is referenced by: dvn2bss 25192 dvnres 25193 cpnord 25197 taylfvallem1 25614 tayl0 25619 taylply2 25625 taylply 25626 dvtaylp 25627 dvntaylp 25628 dvntaylp0 25629 taylthlem1 25630 taylthlem2 25631

W3C validator