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Theorem lcmineqlem8 42489

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 11122	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
3		1cnd 11130	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
4		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5	3, 4	subcld 11496	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
6		neg1cn 12135	. . . 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 12541	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		lcmineqlem8.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	12	nnzd 12541	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
14		znnsub 12564	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
15	11, 13, 14	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
16	9, 15	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ)
17	16	nnnn0d 12489	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℕ₀)
19	8, 18	expcld 14099	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 𝑀)) ∈ ℂ)
20	12	nncnd 12181	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
22	10	nncnd 12181	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
24	21, 23	subcld 11496	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
25		nnm1nn0 12469	. . . . . . 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 14032	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ ((𝑁 − 𝑀) − 1) ∈ ℕ₀) → (𝑦↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
29	8, 27, 28	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
30	24, 29	mulcld 11156	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) ∈ ℂ)
31		lcmineqlem7 42488	. . . 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 25930	. . . 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 7367	. . 3 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
36		oveq1 7367	. . . 4 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑((𝑁 − 𝑀) − 1)) = ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))
37	36	oveq2d 7376	. . 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 25949	. 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 11496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ)
42		ax-1cn 11087	. . . . . . . 8 ⊢ 1 ∈ ℂ
43		subcl 11383	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ)
44	42, 43	mpan 691	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 − 𝑥) ∈ ℂ)
45		expcl 14032	. . . . . . 7 ⊢ (((1 − 𝑥) ∈ ℂ ∧ ((𝑁 − 𝑀) − 1) ∈ ℕ₀) → ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
46	44, 26, 45	syl2anr 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)) ∈ ℂ)
47	41, 46, 7	mul32d 11347	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
48	20, 22	subcld 11496	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
49	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
50	48, 49	mulcomd 11157	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑀) · -1) = (-1 · (𝑁 − 𝑀)))
51	50	oveq1d 7375	. . . . . 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 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))
54	48	mulm1d 11593	. . . . . 6 ⊢ (𝜑 → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
55	54	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀))
56	55	oveq1d 7375	. . . 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 5179	. 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 (class class class)co 7360 ℂcc 11027 ℝcr 11028 1c1 11030 · cmul 11034 < clt 11170 − cmin 11368 -cneg 11369 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ↑cexp 14014 D cdv 25840

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844

This theorem is referenced by: lcmineqlem10 42491

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