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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem3 | Structured version Visualization version GIF version |
Description: Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 30-Apr-2024.) |
Ref | Expression |
---|---|
lcmineqlem3.1 | ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 |
lcmineqlem3.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
lcmineqlem3.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
lcmineqlem3.4 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
Ref | Expression |
---|---|
lcmineqlem3 | ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmineqlem3.1 | . . 3 ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 | |
2 | lcmineqlem3.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | lcmineqlem3.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
4 | lcmineqlem3.4 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | |
5 | 1, 2, 3, 4 | lcmineqlem2 39182 | . 2 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
6 | elunitcn 12837 | . . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ) | |
7 | 6 | 3ad2ant3 1131 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ℂ) |
8 | elfznn0 12984 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) | |
9 | 8 | 3ad2ant2 1130 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑘 ∈ ℕ0) |
10 | nnm1nn0 11917 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑀 − 1) ∈ ℕ0) |
13 | 7, 9, 12 | expaddd 13497 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
14 | 13 | 3expa 1114 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
15 | 14 | itgeq2dv 24364 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥) |
16 | 15 | oveq2d 7149 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
17 | 16 | sumeq2dv 15040 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
18 | 0red 10622 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ∈ ℝ) | |
19 | 1red 10620 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℝ) | |
20 | 0le1 11141 | . . . . . . 7 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ≤ 1) |
22 | 11 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 − 1) ∈ ℕ0) |
23 | 8 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
24 | 22, 23 | nn0addcld 11938 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑀 − 1) + 𝑘) ∈ ℕ0) |
25 | 18, 19, 21, 24 | itgpowd 24632 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) / (((𝑀 − 1) + 𝑘) + 1))) |
26 | 3 | nncnd 11632 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
27 | 26 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℂ) |
28 | 1cnd 10614 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℂ) | |
29 | nn0cn 11886 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
30 | 8, 29 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℂ) |
31 | 30 | adantl 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℂ) |
32 | 27, 28, 31 | nppcand 11000 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑀 − 1) + 𝑘) + 1) = (𝑀 + 𝑘)) |
33 | 32 | oveq2d 7149 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(((𝑀 − 1) + 𝑘) + 1)) = (1↑(𝑀 + 𝑘))) |
34 | 32 | oveq2d 7149 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(((𝑀 − 1) + 𝑘) + 1)) = (0↑(𝑀 + 𝑘))) |
35 | 33, 34 | oveq12d 7151 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘)))) |
36 | 3 | adantr 483 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℕ) |
37 | nnnn0addcl 11906 | . . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑀 + 𝑘) ∈ ℕ) | |
38 | 36, 23, 37 | syl2anc 586 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℕ) |
39 | 38 | nnzd 12065 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℤ) |
40 | 1exp 13443 | . . . . . . . . . 10 ⊢ ((𝑀 + 𝑘) ∈ ℤ → (1↑(𝑀 + 𝑘)) = 1) | |
41 | 39, 40 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(𝑀 + 𝑘)) = 1) |
42 | 0exp 13449 | . . . . . . . . . 10 ⊢ ((𝑀 + 𝑘) ∈ ℕ → (0↑(𝑀 + 𝑘)) = 0) | |
43 | 38, 42 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(𝑀 + 𝑘)) = 0) |
44 | 41, 43 | oveq12d 7151 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘))) = (1 − 0)) |
45 | 35, 44 | eqtrd 2855 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = (1 − 0)) |
46 | 1m0e1 11737 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
47 | 45, 46 | syl6eq 2871 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = 1) |
48 | 47, 32 | oveq12d 7151 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) / (((𝑀 − 1) + 𝑘) + 1)) = (1 / (𝑀 + 𝑘))) |
49 | 25, 48 | eqtrd 2855 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (1 / (𝑀 + 𝑘))) |
50 | 49 | oveq2d 7149 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) |
51 | 50 | sumeq2dv 15040 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) |
52 | 5, 17, 51 | 3eqtr2d 2861 | 1 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5042 (class class class)co 7133 ℂcc 10513 0cc0 10515 1c1 10516 + caddc 10518 · cmul 10520 ≤ cle 10654 − cmin 10848 -cneg 10849 / cdiv 11275 ℕcn 11616 ℕ0cn0 11876 ℤcz 11960 [,]cicc 12720 ...cfz 12876 ↑cexp 13414 Ccbc 13647 Σcsu 15022 ∫citg 24201 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5166 ax-sep 5179 ax-nul 5186 ax-pow 5242 ax-pr 5306 ax-un 7439 ax-inf2 9082 ax-cc 9835 ax-cnex 10571 ax-resscn 10572 ax-1cn 10573 ax-icn 10574 ax-addcl 10575 ax-addrcl 10576 ax-mulcl 10577 ax-mulrcl 10578 ax-mulcom 10579 ax-addass 10580 ax-mulass 10581 ax-distr 10582 ax-i2m1 10583 ax-1ne0 10584 ax-1rid 10585 ax-rnegex 10586 ax-rrecex 10587 ax-cnre 10588 ax-pre-lttri 10589 ax-pre-lttrn 10590 ax-pre-ltadd 10591 ax-pre-mulgt0 10592 ax-pre-sup 10593 ax-addf 10594 ax-mulf 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3007 df-nel 3111 df-ral 3130 df-rex 3131 df-reu 3132 df-rmo 3133 df-rab 3134 df-v 3475 df-sbc 3753 df-csb 3861 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-symdif 4197 df-nul 4270 df-if 4444 df-pw 4517 df-sn 4544 df-pr 4546 df-tp 4548 df-op 4550 df-uni 4815 df-int 4853 df-iun 4897 df-iin 4898 df-disj 5008 df-br 5043 df-opab 5105 df-mpt 5123 df-tr 5149 df-id 5436 df-eprel 5441 df-po 5450 df-so 5451 df-fr 5490 df-se 5491 df-we 5492 df-xp 5537 df-rel 5538 df-cnv 5539 df-co 5540 df-dm 5541 df-rn 5542 df-res 5543 df-ima 5544 df-pred 6124 df-ord 6170 df-on 6171 df-lim 6172 df-suc 6173 df-iota 6290 df-fun 6333 df-fn 6334 df-f 6335 df-f1 6336 df-fo 6337 df-f1o 6338 df-fv 6339 df-isom 6340 df-riota 7091 df-ov 7136 df-oprab 7137 df-mpo 7138 df-of 7387 df-ofr 7388 df-om 7559 df-1st 7667 df-2nd 7668 df-supp 7809 df-wrecs 7925 df-recs 7986 df-rdg 8024 df-1o 8080 df-2o 8081 df-oadd 8084 df-omul 8085 df-er 8267 df-map 8386 df-pm 8387 df-ixp 8440 df-en 8488 df-dom 8489 df-sdom 8490 df-fin 8491 df-fsupp 8812 df-fi 8853 df-sup 8884 df-inf 8885 df-oi 8952 df-dju 9308 df-card 9346 df-acn 9349 df-pnf 10655 df-mnf 10656 df-xr 10657 df-ltxr 10658 df-le 10659 df-sub 10850 df-neg 10851 df-div 11276 df-nn 11617 df-2 11679 df-3 11680 df-4 11681 df-5 11682 df-6 11683 df-7 11684 df-8 11685 df-9 11686 df-n0 11877 df-z 11961 df-dec 12078 df-uz 12223 df-q 12328 df-rp 12369 df-xneg 12486 df-xadd 12487 df-xmul 12488 df-ioo 12721 df-ioc 12722 df-ico 12723 df-icc 12724 df-fz 12877 df-fzo 13018 df-fl 13146 df-mod 13222 df-seq 13354 df-exp 13415 df-fac 13619 df-bc 13648 df-hash 13676 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-limsup 14808 df-clim 14825 df-rlim 14826 df-sum 15023 df-struct 16464 df-ndx 16465 df-slot 16466 df-base 16468 df-sets 16469 df-ress 16470 df-plusg 16557 df-mulr 16558 df-starv 16559 df-sca 16560 df-vsca 16561 df-ip 16562 df-tset 16563 df-ple 16564 df-ds 16566 df-unif 16567 df-hom 16568 df-cco 16569 df-rest 16675 df-topn 16676 df-0g 16694 df-gsum 16695 df-topgen 16696 df-pt 16697 df-prds 16700 df-xrs 16754 df-qtop 16759 df-imas 16760 df-xps 16762 df-mre 16836 df-mrc 16837 df-acs 16839 df-mgm 17831 df-sgrp 17880 df-mnd 17891 df-submnd 17936 df-mulg 18204 df-cntz 18426 df-cmn 18887 df-psmet 20513 df-xmet 20514 df-met 20515 df-bl 20516 df-mopn 20517 df-fbas 20518 df-fg 20519 df-cnfld 20522 df-top 21478 df-topon 21495 df-topsp 21517 df-bases 21530 df-cld 21603 df-ntr 21604 df-cls 21605 df-nei 21682 df-lp 21720 df-perf 21721 df-cn 21811 df-cnp 21812 df-haus 21899 df-cmp 21971 df-tx 22146 df-hmeo 22339 df-fil 22430 df-fm 22522 df-flim 22523 df-flf 22524 df-xms 22906 df-ms 22907 df-tms 22908 df-cncf 23462 df-ovol 24047 df-vol 24048 df-mbf 24202 df-itg1 24203 df-itg2 24204 df-ibl 24205 df-itg 24206 df-0p 24253 df-limc 24448 df-dv 24449 |
This theorem is referenced by: lcmineqlem6 39186 |
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