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Theorem lcmineqlem8 40889

Description: Derivative of (1-x)^(N-M). (Contributed by metakunt, 12-May-2024.)

**Hypotheses**
Ref	Expression
lcmineqlem8.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
lcmineqlem8.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
lcmineqlem8.3	⊢ (𝜑 → 𝑀 < 𝑁)

**Assertion**
Ref	Expression
lcmineqlem8	⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))))

Distinct variable groups: 𝑥,𝑀 𝑥,𝑁 𝜑,𝑥

Proof of Theorem lcmineqlem8 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnelprrecn 11199	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
3		1cnd 11205	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
4		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5	3, 4	subcld 11567	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
6		neg1cn 12322	. . . 4 ⊢ -1 ∈ ℂ
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → -1 ∈ ℂ)
8		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
9		lcmineqlem8.3	. . . . . . 7 ⊢ (𝜑 → 𝑀 < 𝑁)
10		lcmineqlem8.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
11	10	nnzd 12581	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		lcmineqlem8.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nnzd 12581	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
14		znnsub 12604	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
15	11, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
16	9, 15	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ)
17	16	nnnn0d 12528	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
18	17	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℕ₀)
19	8, 18	expcld 14107	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 𝑀)) ∈ ℂ)
20	12	nncnd 12224	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
22	10	nncnd 12224	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
23	22	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
24	21, 23	subcld 11567	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
25		nnm1nn0 12509	. . . . . . 7 ⊢ ((𝑁 − 𝑀) ∈ ℕ → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
26	16, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
27	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
28		expcl 14041	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ ((𝑁 − 𝑀) − 1) ∈ ℕ₀) → (𝑦↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
29	8, 27, 28	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
30	24, 29	mulcld 11230	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) ∈ ℂ)
31		lcmineqlem7 40888	. . . 4 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (1 − 𝑥))) = (𝑥 ∈ ℂ ↦ -1)
32	31	a1i 11	. . 3 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (1 − 𝑥))) = (𝑥 ∈ ℂ ↦ -1))
33		dvexp 25461	. . . 4 ⊢ ((𝑁 − 𝑀) ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑(𝑁 − 𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1)))))
34	16, 33	syl 17	. . 3 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑(𝑁 − 𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1)))))
35		oveq1 7412	. . 3 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
36		oveq1 7412	. . . 4 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑((𝑁 − 𝑀) − 1)) = ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))
37	36	oveq2d 7421	. . 3 ⊢ (𝑦 = (1 − 𝑥) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) = ((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
38	2, 2, 5, 7, 19, 30, 32, 34, 35, 37	dvmptco 25480	. 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1)))
39	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℂ)
40	22	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑀 ∈ ℂ)
41	39, 40	subcld 11567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
42		ax-1cn 11164	. . . . . . . 8 ⊢ 1 ∈ ℂ
43		subcl 11455	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
44	42, 43	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 − 𝑥) ∈ ℂ)
45		expcl 14041	. . . . . . 7 ⊢ (((1 − 𝑥) ∈ ℂ ∧ ((𝑁 − 𝑀) − 1) ∈ ℕ₀) → ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
46	44, 26, 45	syl2anr 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
47	41, 46, 7	mul32d 11420	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
48	20, 22	subcld 11567	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
49	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
50	48, 49	mulcomd 11231	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑀) · -1) = (-1 · (𝑁 − 𝑀)))
51	50	oveq1d 7420	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
52	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
53	47, 52	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
54	48	mulm1d 11662	. . . . . 6 ⊢ (𝜑 → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
55	54	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
56	55	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
57	53, 56	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
58	57	mpteq2dva 5247	. 2 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1)) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))))
59	38, 58	eqtrd 2772	1 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cpr 4629 class class class wbr 5147 ↦ cmpt 5230 (class class class)co 7405 ℂcc 11104 ℝcr 11105 1c1 11107 · cmul 11111 < clt 11244 − cmin 11440 -cneg 11441 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ↑cexp 14023 D cdv 25371

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375

This theorem is referenced by: lcmineqlem10 40891

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