dvxpaek - Mathbox for Glauco Siliprandi

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Theorem dvxpaek 45986

Description: Derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvxpaek.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvxpaek.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
dvxpaek.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
dvxpaek.k	⊢ (𝜑 → 𝐾 ∈ ℕ)

**Assertion**
Ref	Expression
dvxpaek	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐾 𝑥,𝑆 𝑥,𝑋 𝜑,𝑥

Proof of Theorem dvxpaek Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvxpaek.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		cnelprrecn 11099	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
4		dvxpaek.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
5	1, 4	dvdmsscn 45982	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ ℂ)
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8	6, 7	sseldd 3930	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
9		dvxpaek.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
11	8, 10	addcld 11131	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 + 𝐴) ∈ ℂ)
12		1red 11113	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
13		0red 11115	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
14	12, 13	readdcld 11141	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 + 0) ∈ ℝ)
15		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
16		dvxpaek.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
17	16	nnnn0d 12442	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℕ₀)
19	15, 18	expcld 14053	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝐾) ∈ ℂ)
20	18	nn0cnd 12444	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℂ)
21		nnm1nn0 12422	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ₀)
22	16, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ₀)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 − 1) ∈ ℕ₀)
24	15, 23	expcld 14053	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝐾 − 1)) ∈ ℂ)
25	20, 24	mulcld 11132	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 · (𝑦↑(𝐾 − 1))) ∈ ℂ)
26	1, 4	dvmptidg 45963	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝑥)) = (𝑥 ∈ 𝑋 ↦ 1))
27	1, 4, 9	dvmptconst 45961	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
28	1, 8, 12, 26, 10, 13, 27	dvmptadd 25891	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑥 + 𝐴))) = (𝑥 ∈ 𝑋 ↦ (1 + 0)))
29		dvexp 25884	. . . 4 ⊢ (𝐾 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝐾))) = (𝑦 ∈ ℂ ↦ (𝐾 · (𝑦↑(𝐾 − 1)))))
30	16, 29	syl 17	. . 3 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝐾))) = (𝑦 ∈ ℂ ↦ (𝐾 · (𝑦↑(𝐾 − 1)))))
31		oveq1 7353	. . 3 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑𝐾) = ((𝑥 + 𝐴)↑𝐾))
32		oveq1 7353	. . . 4 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑(𝐾 − 1)) = ((𝑥 + 𝐴)↑(𝐾 − 1)))
33	32	oveq2d 7362	. . 3 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝐾 · (𝑦↑(𝐾 − 1))) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
34	1, 3, 11, 14, 19, 25, 28, 30, 31, 33	dvmptco 25903	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))))
35		1p0e1 12244	. . . . . 6 ⊢ (1 + 0) = 1
36	35	oveq2i 7357	. . . . 5 ⊢ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1)
37	36	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1))
38	16	nncnd 12141	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℂ)
40	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 − 1) ∈ ℕ₀)
41	11, 40	expcld 14053	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 + 𝐴)↑(𝐾 − 1)) ∈ ℂ)
42	39, 41	mulcld 11132	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) ∈ ℂ)
43	42	mulridd 11129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
44	37, 43	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
45	44	mpteq2dva 5182	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
46	34, 45	eqtrd 2766	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⊆ wss 3897 {cpr 4575 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 − cmin 11344 ℕcn 12125 ℕ₀cn0 12381 ↑cexp 13968 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21291 D cdv 25791

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 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 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795

This theorem is referenced by: dvnxpaek 45988

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