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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem8 | Structured version Visualization version GIF version |
Description: Derivative of (1-x)^(N-M). (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
lcmineqlem8.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
lcmineqlem8.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
lcmineqlem8.3 | ⊢ (𝜑 → 𝑀 < 𝑁) |
Ref | Expression |
---|---|
lcmineqlem8 | ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnelprrecn 11066 | . . . 4 ⊢ ℂ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ}) |
3 | 1cnd 11072 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ) | |
4 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
5 | 3, 4 | subcld 11434 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ) |
6 | neg1cn 12189 | . . . 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 12527 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | lcmineqlem8.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
13 | 12 | nnzd 12527 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
14 | znnsub 12468 | . . . . . . . 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 12395 | . . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ0) |
18 | 17 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℕ0) |
19 | 8, 18 | expcld 13966 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 𝑀)) ∈ ℂ) |
20 | 12 | nncnd 12091 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
21 | 20 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ) |
22 | 10 | nncnd 12091 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ) |
24 | 21, 23 | subcld 11434 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ) |
25 | nnm1nn0 12376 | . . . . . . 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 13902 | . . . . 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 11097 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑀) · (𝑦↑((𝑁 − 𝑀) − 1))) ∈ ℂ) |
31 | lcmineqlem7 40348 | . . . 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 25224 | . . . 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 7345 | . . 3 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀))) | |
36 | oveq1 7345 | . . . 4 ⊢ (𝑦 = (1 − 𝑥) → (𝑦↑((𝑁 − 𝑀) − 1)) = ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) | |
37 | 36 | oveq2d 7354 | . . 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 25243 | . 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 11434 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑁 − 𝑀) ∈ ℂ) |
42 | ax-1cn 11031 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
43 | subcl 11322 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 − 𝑥) ∈ ℂ) | |
44 | 42, 43 | mpan 687 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (1 − 𝑥) ∈ ℂ) |
45 | expcl 13902 | . . . . . . 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 11287 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (((𝑁 − 𝑀) · -1) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))) |
48 | 20, 22 | subcld 11434 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ) |
49 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ) |
50 | 48, 49 | mulcomd 11098 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑀) · -1) = (-1 · (𝑁 − 𝑀))) |
51 | 50 | oveq1d 7353 | . . . . . 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 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))) |
54 | 48 | mulm1d 11529 | . . . . . 6 ⊢ (𝜑 → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀)) |
55 | 54 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (-1 · (𝑁 − 𝑀)) = -(𝑁 − 𝑀)) |
56 | 55 | oveq1d 7353 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((-1 · (𝑁 − 𝑀)) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))) |
57 | 53, 56 | eqtrd 2776 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))) · -1) = (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))) |
58 | 57 | mpteq2dva 5193 | . 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 2776 | 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 1540 ∈ wcel 2105 {cpr 4576 class class class wbr 5093 ↦ cmpt 5176 (class class class)co 7338 ℂcc 10971 ℝcr 10972 1c1 10974 · cmul 10978 < clt 11111 − cmin 11307 -cneg 11308 ℕcn 12075 ℕ0cn0 12335 ℤcz 12421 ↑cexp 13884 D cdv 25134 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-pre-sup 11051 ax-addf 11052 ax-mulf 11053 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-iin 4945 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-se 5577 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-isom 6489 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-of 7596 df-om 7782 df-1st 7900 df-2nd 7901 df-supp 8049 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-2o 8369 df-er 8570 df-map 8689 df-pm 8690 df-ixp 8758 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-fsupp 9228 df-fi 9269 df-sup 9300 df-inf 9301 df-oi 9368 df-card 9797 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 df-nn 12076 df-2 12138 df-3 12139 df-4 12140 df-5 12141 df-6 12142 df-7 12143 df-8 12144 df-9 12145 df-n0 12336 df-z 12422 df-dec 12540 df-uz 12685 df-q 12791 df-rp 12833 df-xneg 12950 df-xadd 12951 df-xmul 12952 df-icc 13188 df-fz 13342 df-fzo 13485 df-seq 13824 df-exp 13885 df-hash 14147 df-cj 14910 df-re 14911 df-im 14912 df-sqrt 15046 df-abs 15047 df-struct 16946 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-mulr 17074 df-starv 17075 df-sca 17076 df-vsca 17077 df-ip 17078 df-tset 17079 df-ple 17080 df-ds 17082 df-unif 17083 df-hom 17084 df-cco 17085 df-rest 17231 df-topn 17232 df-0g 17250 df-gsum 17251 df-topgen 17252 df-pt 17253 df-prds 17256 df-xrs 17311 df-qtop 17316 df-imas 17317 df-xps 17319 df-mre 17393 df-mrc 17394 df-acs 17396 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-submnd 18529 df-mulg 18798 df-cntz 19020 df-cmn 19484 df-psmet 20696 df-xmet 20697 df-met 20698 df-bl 20699 df-mopn 20700 df-fbas 20701 df-fg 20702 df-cnfld 20705 df-top 22150 df-topon 22167 df-topsp 22189 df-bases 22203 df-cld 22277 df-ntr 22278 df-cls 22279 df-nei 22356 df-lp 22394 df-perf 22395 df-cn 22485 df-cnp 22486 df-haus 22573 df-tx 22820 df-hmeo 23013 df-fil 23104 df-fm 23196 df-flim 23197 df-flf 23198 df-xms 23580 df-ms 23581 df-tms 23582 df-cncf 24148 df-limc 25137 df-dv 25138 |
This theorem is referenced by: lcmineqlem10 40351 |
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