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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 40038 | . 2 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
6 | elunitcn 13200 | . . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ) | |
7 | 6 | 3ad2ant3 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ℂ) |
8 | elfznn0 13349 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) | |
9 | 8 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑘 ∈ ℕ0) |
10 | nnm1nn0 12274 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | 3ad2ant1 1132 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑀 − 1) ∈ ℕ0) |
13 | 7, 9, 12 | expaddd 13866 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
14 | 13 | 3expa 1117 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
15 | 14 | itgeq2dv 24946 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥) |
16 | 15 | oveq2d 7291 | . . 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 15415 | . 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 10978 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ∈ ℝ) | |
19 | 1red 10976 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℝ) | |
20 | 0le1 11498 | . . . . . . 7 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ≤ 1) |
22 | 11 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 − 1) ∈ ℕ0) |
23 | 8 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
24 | 22, 23 | nn0addcld 12297 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑀 − 1) + 𝑘) ∈ ℕ0) |
25 | 18, 19, 21, 24 | itgpowd 25214 | . . . . 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 11989 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
27 | 26 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℂ) |
28 | 1cnd 10970 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℂ) | |
29 | nn0cn 12243 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
30 | 8, 29 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℂ) |
31 | 30 | adantl 482 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℂ) |
32 | 27, 28, 31 | nppcand 11357 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑀 − 1) + 𝑘) + 1) = (𝑀 + 𝑘)) |
33 | 32 | oveq2d 7291 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(((𝑀 − 1) + 𝑘) + 1)) = (1↑(𝑀 + 𝑘))) |
34 | 32 | oveq2d 7291 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(((𝑀 − 1) + 𝑘) + 1)) = (0↑(𝑀 + 𝑘))) |
35 | 33, 34 | oveq12d 7293 | . . . . . . . 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 481 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℕ) |
37 | nnnn0addcl 12263 | . . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑀 + 𝑘) ∈ ℕ) | |
38 | 36, 23, 37 | syl2anc 584 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℕ) |
39 | 38 | nnzd 12425 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℤ) |
40 | 1exp 13812 | . . . . . . . . . 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 13818 | . . . . . . . . . 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 7293 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘))) = (1 − 0)) |
45 | 35, 44 | eqtrd 2778 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = (1 − 0)) |
46 | 1m0e1 12094 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
47 | 45, 46 | eqtrdi 2794 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = 1) |
48 | 47, 32 | oveq12d 7293 | . . . . 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 2778 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (1 / (𝑀 + 𝑘))) |
50 | 49 | oveq2d 7291 | . . 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 15415 | . 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 2784 | 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 ℕ0cn0 12233 ℤcz 12319 [,]cicc 13082 ...cfz 13239 ↑cexp 13782 Ccbc 14016 Σcsu 15397 ∫citg 24782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-limc 25030 df-dv 25031 |
This theorem is referenced by: lcmineqlem6 40042 |
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