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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 39318 | . 2 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
6 | elunitcn 12846 | . . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ) | |
7 | 6 | 3ad2ant3 1132 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ℂ) |
8 | elfznn0 12995 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) | |
9 | 8 | 3ad2ant2 1131 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑘 ∈ ℕ0) |
10 | nnm1nn0 11926 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | 3ad2ant1 1130 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑀 − 1) ∈ ℕ0) |
13 | 7, 9, 12 | expaddd 13508 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
14 | 13 | 3expa 1115 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
15 | 14 | itgeq2dv 24385 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥) |
16 | 15 | oveq2d 7151 | . . 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 15052 | . 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 10633 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ∈ ℝ) | |
19 | 1red 10631 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℝ) | |
20 | 0le1 11152 | . . . . . . 7 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ≤ 1) |
22 | 11 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 − 1) ∈ ℕ0) |
23 | 8 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
24 | 22, 23 | nn0addcld 11947 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑀 − 1) + 𝑘) ∈ ℕ0) |
25 | 18, 19, 21, 24 | itgpowd 24653 | . . . . 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 11641 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
27 | 26 | adantr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℂ) |
28 | 1cnd 10625 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℂ) | |
29 | nn0cn 11895 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
30 | 8, 29 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℂ) |
31 | 30 | adantl 485 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℂ) |
32 | 27, 28, 31 | nppcand 11011 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑀 − 1) + 𝑘) + 1) = (𝑀 + 𝑘)) |
33 | 32 | oveq2d 7151 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(((𝑀 − 1) + 𝑘) + 1)) = (1↑(𝑀 + 𝑘))) |
34 | 32 | oveq2d 7151 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(((𝑀 − 1) + 𝑘) + 1)) = (0↑(𝑀 + 𝑘))) |
35 | 33, 34 | oveq12d 7153 | . . . . . . . 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 484 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℕ) |
37 | nnnn0addcl 11915 | . . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑀 + 𝑘) ∈ ℕ) | |
38 | 36, 23, 37 | syl2anc 587 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℕ) |
39 | 38 | nnzd 12074 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℤ) |
40 | 1exp 13454 | . . . . . . . . . 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 13460 | . . . . . . . . . 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 7153 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘))) = (1 − 0)) |
45 | 35, 44 | eqtrd 2833 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = (1 − 0)) |
46 | 1m0e1 11746 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
47 | 45, 46 | eqtrdi 2849 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = 1) |
48 | 47, 32 | oveq12d 7153 | . . . . 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 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (1 / (𝑀 + 𝑘))) |
50 | 49 | oveq2d 7151 | . . 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 15052 | . 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 2839 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 [,]cicc 12729 ...cfz 12885 ↑cexp 13425 Ccbc 13658 Σcsu 15034 ∫citg 24222 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-limc 24469 df-dv 24470 |
This theorem is referenced by: lcmineqlem6 39322 |
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