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Theorem lcmineqlem8 42150

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 11106	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
3		1cnd 11114	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
4		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5	3, 4	subcld 11479	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
6		neg1cn 12117	. . . 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 12501	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		lcmineqlem8.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nnzd 12501	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
14		znnsub 12524	. . . . . . . 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 12449	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℕ₀)
19	8, 18	expcld 14055	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 𝑀)) ∈ ℂ)
20	12	nncnd 12148	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
22	10	nncnd 12148	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
24	21, 23	subcld 11479	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
25		nnm1nn0 12429	. . . . . . 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 13988	. . . . 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 11139	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) ∈ ℂ)
31		lcmineqlem7 42149	. . . 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 25885	. . . 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 7359	. . 3 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
36		oveq1 7359	. . . 4 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑((𝑁 − 𝑀) − 1)) = ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))
37	36	oveq2d 7368	. . 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 25904	. 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 11479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
42		ax-1cn 11071	. . . . . . . 8 ⊢ 1 ∈ ℂ
43		subcl 11366	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
44	42, 43	mpan 690	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 − 𝑥) ∈ ℂ)
45		expcl 13988	. . . . . . 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 11330	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
48	20, 22	subcld 11479	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
49	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
50	48, 49	mulcomd 11140	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑀) · -1) = (-1 · (𝑁 − 𝑀)))
51	50	oveq1d 7367	. . . . . 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 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
54	48	mulm1d 11576	. . . . . 6 ⊢ (𝜑 → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
55	54	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
56	55	oveq1d 7367	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
57	53, 56	eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
58	57	mpteq2dva 5186	. 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 2768	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 1541 ∈ wcel 2113 {cpr 4577 class class class wbr 5093 ↦ cmpt 5174 (class class class)co 7352 ℂcc 11011 ℝcr 11012 1c1 11014 · cmul 11018 < clt 11153 − cmin 11351 -cneg 11352 ℕcn 12132 ℕ₀cn0 12388 ℤcz 12475 ↑cexp 13970 D cdv 25792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796

This theorem is referenced by: lcmineqlem10 42152

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