dvxpaek - Mathbox for Glauco Siliprandi

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Theorem dvxpaek 46514

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 11166	. . . 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 46510	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ ℂ)
7		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8	6, 7	sseldd 3937	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
9		dvxpaek.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
11	8, 10	addcld 11201	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 + 𝐴) ∈ ℂ)
12		1red 11182	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
13		0red 11184	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
14	12, 13	readdcld 11211	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 + 0) ∈ ℝ)
15		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
16		dvxpaek.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
17	16	nnnn0d 12542	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
18	17	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℕ₀)
19	15, 18	expcld 14159	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝐾) ∈ ℂ)
20	18	nn0cnd 12544	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐾 ∈ ℂ)
21		nnm1nn0 12522	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ₀)
22	16, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ₀)
23	22	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 − 1) ∈ ℕ₀)
24	15, 23	expcld 14159	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝐾 − 1)) ∈ ℂ)
25	20, 24	mulcld 11202	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐾 · (𝑦↑(𝐾 − 1))) ∈ ℂ)
26	1, 4	dvmptidg 46491	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝑥)) = (𝑥 ∈ 𝑋 ↦ 1))
27	1, 4, 9	dvmptconst 46489	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
28	1, 8, 12, 26, 10, 13, 27	dvmptadd 26022	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑥 + 𝐴))) = (𝑥 ∈ 𝑋 ↦ (1 + 0)))
29		dvexp 26015	. . . 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 7403	. . 3 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑𝐾) = ((𝑥 + 𝐴)↑𝐾))
32		oveq1 7403	. . . 4 ⊢ (𝑦 = (𝑥 + 𝐴) → (𝑦↑(𝐾 − 1)) = ((𝑥 + 𝐴)↑(𝐾 − 1)))
33	32	oveq2d 7412	. . 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 26034	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))))
35		1p0e1 12340	. . . . . 6 ⊢ (1 + 0) = 1
36	35	oveq2i 7407	. . . . 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 12226	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
39	38	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℂ)
40	22	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 − 1) ∈ ℕ₀)
41	11, 40	expcld 14159	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 + 𝐴)↑(𝐾 − 1)) ∈ ℂ)
42	39, 41	mulcld 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) ∈ ℂ)
43	42	mulridd 11199	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · 1) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
44	37, 43	eqtrd 2797	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0)) = (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))))
45	44	mpteq2dva 5193	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1))) · (1 + 0))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
46	34, 45	eqtrd 2797	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 399 = wceq 1560 ∈ wcel 2142 ⊆ wss 3904 {cpr 4584 ↦ cmpt 5181 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 · cmul 11078 − cmin 11414 ℕcn 12210 ℕ₀cn0 12481 ↑cexp 14074 ↾_t crest 17449 TopOpenctopn 17450 ℂ_fldccnfld 21424 D cdv 25925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-icc 13356 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-limc 25928 df-dv 25929

This theorem is referenced by: dvnxpaek 46516

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