dvxpaek - Mathbox for Glauco Siliprandi

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Theorem dvxpaek 45861

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 11277	. . . 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 45857	. . . . . 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 4009	. . . 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 11309	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 + 𝐴) ∈ ℂ)
12		1red 11291	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
13		0red 11293	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
14	12, 13	readdcld 11319	. . 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 12613	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℕ₀)
19	15, 18	expcld 14196	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝐾) ∈ ℂ)
20	18	nn0cnd 12615	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℂ)
21		nnm1nn0 12594	. . . . . . 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 14196	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝐾 − 1)) ∈ ℂ)
25	20, 24	mulcld 11310	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 · (𝑦↑(𝐾 − 1))) ∈ ℂ)
26	1, 4	dvmptidg 45838	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝑥)) = (𝑥 ∈ 𝑋 ↦ 1))
27	1, 4, 9	dvmptconst 45836	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
28	1, 8, 12, 26, 10, 13, 27	dvmptadd 26018	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑥 + 𝐴))) = (𝑥 ∈ 𝑋 ↦ (1 + 0)))
29		dvexp 26011	. . . 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 7455	. . 3 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑𝐾) = ((𝑥 + 𝐴)↑𝐾))
32		oveq1 7455	. . . 4 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑(𝐾 − 1)) = ((𝑥 + 𝐴)↑(𝐾 − 1)))
33	32	oveq2d 7464	. . 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 26030	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))))
35		1p0e1 12417	. . . . . 6 ⊢ (1 + 0) = 1
36	35	oveq2i 7459	. . . . 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 12309	. . . . . . 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 14196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 + 𝐴)↑(𝐾 − 1)) ∈ ℂ)
42	39, 41	mulcld 11310	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) ∈ ℂ)
43	42	mulridd 11307	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
44	37, 43	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
45	44	mpteq2dva 5266	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
46	34, 45	eqtrd 2780	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 1537 ∈ wcel 2108 ⊆ wss 3976 {cpr 4650 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 ℕcn 12293 ℕ₀cn0 12553 ↑cexp 14112 ↾_t crest 17480 TopOpenctopn 17481 ℂ_fldccnfld 21387 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: dvnxpaek 45863

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