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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 40487 | . 2 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
6 | elunitcn 13385 | . . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ) | |
7 | 6 | 3ad2ant3 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ℂ) |
8 | elfznn0 13534 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) | |
9 | 8 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑘 ∈ ℕ0) |
10 | nnm1nn0 12454 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑀 − 1) ∈ ℕ0) |
13 | 7, 9, 12 | expaddd 14053 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
14 | 13 | 3expa 1118 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
15 | 14 | itgeq2dv 25146 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥) |
16 | 15 | oveq2d 7373 | . . 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 15588 | . 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 11158 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ∈ ℝ) | |
19 | 1red 11156 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℝ) | |
20 | 0le1 11678 | . . . . . . 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 12477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑀 − 1) + 𝑘) ∈ ℕ0) |
25 | 18, 19, 21, 24 | itgpowd 25414 | . . . . 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 12169 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
27 | 26 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℂ) |
28 | 1cnd 11150 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℂ) | |
29 | nn0cn 12423 | . . . . . . . . . . . . 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 11537 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑀 − 1) + 𝑘) + 1) = (𝑀 + 𝑘)) |
33 | 32 | oveq2d 7373 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(((𝑀 − 1) + 𝑘) + 1)) = (1↑(𝑀 + 𝑘))) |
34 | 32 | oveq2d 7373 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(((𝑀 − 1) + 𝑘) + 1)) = (0↑(𝑀 + 𝑘))) |
35 | 33, 34 | oveq12d 7375 | . . . . . . . 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 12443 | . . . . . . . . . . . 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 12526 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℤ) |
40 | 1exp 13997 | . . . . . . . . . 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 14003 | . . . . . . . . . 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 7375 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘))) = (1 − 0)) |
45 | 35, 44 | eqtrd 2776 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = (1 − 0)) |
46 | 1m0e1 12274 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
47 | 45, 46 | eqtrdi 2792 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = 1) |
48 | 47, 32 | oveq12d 7375 | . . . . 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 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (1 / (𝑀 + 𝑘))) |
50 | 49 | oveq2d 7373 | . . 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 15588 | . 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 2782 | 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 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 ℕ0cn0 12413 ℤcz 12499 [,]cicc 13267 ...cfz 13424 ↑cexp 13967 Ccbc 14202 Σcsu 15570 ∫citg 24982 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-limc 25230 df-dv 25231 |
This theorem is referenced by: lcmineqlem6 40491 |
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