dvxpaek - Mathbox for Glauco Siliprandi

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Theorem dvxpaek 42215

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 10622	. . . 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 42211	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ ℂ)
7		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8	6, 7	sseldd 3966	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
9		dvxpaek.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
10	9	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
11	8, 10	addcld 10652	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 + 𝐴) ∈ ℂ)
12		1red 10634	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
13		0red 10636	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
14	12, 13	readdcld 10662	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 + 0) ∈ ℝ)
15		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
16		dvxpaek.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
17	16	nnnn0d 11947	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
18	17	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℕ₀)
19	15, 18	expcld 13502	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝐾) ∈ ℂ)
20	18	nn0cnd 11949	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℂ)
21		nnm1nn0 11930	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ₀)
22	16, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ₀)
23	22	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 − 1) ∈ ℕ₀)
24	15, 23	expcld 13502	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝐾 − 1)) ∈ ℂ)
25	20, 24	mulcld 10653	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 · (𝑦↑(𝐾 − 1))) ∈ ℂ)
26	1, 4	dvmptidg 42191	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝑥)) = (𝑥 ∈ 𝑋 ↦ 1))
27	1, 4, 9	dvmptconst 42189	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
28	1, 8, 12, 26, 10, 13, 27	dvmptadd 24549	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑥 + 𝐴))) = (𝑥 ∈ 𝑋 ↦ (1 + 0)))
29		dvexp 24542	. . . 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 7155	. . 3 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑𝐾) = ((𝑥 + 𝐴)↑𝐾))
32		oveq1 7155	. . . 4 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑(𝐾 − 1)) = ((𝑥 + 𝐴)↑(𝐾 − 1)))
33	32	oveq2d 7164	. . 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 24561	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))))
35		1p0e1 11753	. . . . . 6 ⊢ (1 + 0) = 1
36	35	oveq2i 7159	. . . . 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 11646	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
39	38	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℂ)
40	22	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 − 1) ∈ ℕ₀)
41	11, 40	expcld 13502	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 + 𝐴)↑(𝐾 − 1)) ∈ ℂ)
42	39, 41	mulcld 10653	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) ∈ ℂ)
43	42	mulid1d 10650	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
44	37, 43	eqtrd 2854	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
45	44	mpteq2dva 5152	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
46	34, 45	eqtrd 2854	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 398 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 {cpr 4561 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 − cmin 10862 ℕcn 11630 ℕ₀cn0 11889 ↑cexp 13421 ↾_t crest 16686 TopOpenctopn 16687 ℂ_fldccnfld 20537 D cdv 24453

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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-lp 21736 df-perf 21737 df-cn 21827 df-cnp 21828 df-haus 21915 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-limc 24456 df-dv 24457

This theorem is referenced by: dvnxpaek 42217

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