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Theorem lcmineqlem8 42054

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 11227	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
3		1cnd 11235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
4		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5	3, 4	subcld 11599	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
6		neg1cn 12359	. . . 4 ⊢ -1 ∈ ℂ
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → -1 ∈ ℂ)
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
9		lcmineqlem8.3	. . . . . . 7 ⊢ (𝜑 → 𝑀 < 𝑁)
10		lcmineqlem8.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
11	10	nnzd 12620	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		lcmineqlem8.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nnzd 12620	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
14		znnsub 12643	. . . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ)
17	16	nnnn0d 12567	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℕ₀)
19	8, 18	expcld 14169	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 𝑀)) ∈ ℂ)
20	12	nncnd 12261	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
22	10	nncnd 12261	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
24	21, 23	subcld 11599	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
25		nnm1nn0 12547	. . . . . . 7 ⊢ ((𝑁 − 𝑀) ∈ ℕ → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
26	16, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) − 1) ∈ ℕ₀)
28		expcl 14102	. . . . 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 11260	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) ∈ ℂ)
31		lcmineqlem7 42053	. . . 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 25914	. . . 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 7417	. . 3 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
36		oveq1 7417	. . . 4 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑((𝑁 − 𝑀) − 1)) = ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))
37	36	oveq2d 7426	. . 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 25933	. 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1)))
39	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℂ)
40	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑀 ∈ ℂ)
41	39, 40	subcld 11599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
42		ax-1cn 11192	. . . . . . . 8 ⊢ 1 ∈ ℂ
43		subcl 11486	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
44	42, 43	mpan 690	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 − 𝑥) ∈ ℂ)
45		expcl 14102	. . . . . . 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 11450	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
48	20, 22	subcld 11599	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
49	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
50	48, 49	mulcomd 11261	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑀) · -1) = (-1 · (𝑁 − 𝑀)))
51	50	oveq1d 7425	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
52	51	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
53	47, 52	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
54	48	mulm1d 11694	. . . . . 6 ⊢ (𝜑 → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
55	54	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
56	55	oveq1d 7425	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
57	53, 56	eqtrd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
58	57	mpteq2dva 5219	. 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 2771	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cpr 4608 class class class wbr 5124 ↦ cmpt 5206 (class class class)co 7410 ℂcc 11132 ℝcr 11133 1c1 11135 · cmul 11139 < clt 11274 − cmin 11471 -cneg 11472 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ↑cexp 14084 D cdv 25821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-icc 13374 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825

This theorem is referenced by: lcmineqlem10 42056

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