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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 42031 | . 2 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) |
| 6 | elunitcn 13508 | . . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ) | |
| 7 | 6 | 3ad2ant3 1136 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ℂ) |
| 8 | elfznn0 13660 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) | |
| 9 | 8 | 3ad2ant2 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → 𝑘 ∈ ℕ0) |
| 10 | nnm1nn0 12567 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
| 11 | 3, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ0) |
| 12 | 11 | 3ad2ant1 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑀 − 1) ∈ ℕ0) |
| 13 | 7, 9, 12 | expaddd 14188 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀)) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
| 14 | 13 | 3expa 1119 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) ∧ 𝑥 ∈ (0[,]1)) → (𝑥↑((𝑀 − 1) + 𝑘)) = ((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘))) |
| 15 | 14 | itgeq2dv 25817 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥) |
| 16 | 15 | oveq2d 7447 | . . 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 15738 | . 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 11264 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ∈ ℝ) | |
| 19 | 1red 11262 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℝ) | |
| 20 | 0le1 11786 | . . . . . . 7 ⊢ 0 ≤ 1 | |
| 21 | 20 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 0 ≤ 1) |
| 22 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 − 1) ∈ ℕ0) |
| 23 | 8 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
| 24 | 22, 23 | nn0addcld 12591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑀 − 1) + 𝑘) ∈ ℕ0) |
| 25 | 18, 19, 21, 24 | itgpowd 26091 | . . . . 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 12282 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 27 | 26 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℂ) |
| 28 | 1cnd 11256 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 1 ∈ ℂ) | |
| 29 | nn0cn 12536 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ) | |
| 30 | 8, 29 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℂ) |
| 31 | 30 | adantl 481 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℂ) |
| 32 | 27, 28, 31 | nppcand 11645 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑀 − 1) + 𝑘) + 1) = (𝑀 + 𝑘)) |
| 33 | 32 | oveq2d 7447 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑(((𝑀 − 1) + 𝑘) + 1)) = (1↑(𝑀 + 𝑘))) |
| 34 | 32 | oveq2d 7447 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (0↑(((𝑀 − 1) + 𝑘) + 1)) = (0↑(𝑀 + 𝑘))) |
| 35 | 33, 34 | oveq12d 7449 | . . . . . . . 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 480 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑀 ∈ ℕ) |
| 37 | nnnn0addcl 12556 | . . . . . . . . . . . 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 12640 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 𝑘) ∈ ℤ) |
| 40 | 1exp 14132 | . . . . . . . . . 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 14138 | . . . . . . . . . 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 7449 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(𝑀 + 𝑘)) − (0↑(𝑀 + 𝑘))) = (1 − 0)) |
| 45 | 35, 44 | eqtrd 2777 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = (1 − 0)) |
| 46 | 1m0e1 12387 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
| 47 | 45, 46 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑(((𝑀 − 1) + 𝑘) + 1)) − (0↑(((𝑀 − 1) + 𝑘) + 1))) = 1) |
| 48 | 47, 32 | oveq12d 7449 | . . . . 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 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ∫(0[,]1)(𝑥↑((𝑀 − 1) + 𝑘)) d𝑥 = (1 / (𝑀 + 𝑘))) |
| 50 | 49 | oveq2d 7447 | . . 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 15738 | . 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 2783 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 [,]cicc 13390 ...cfz 13547 ↑cexp 14102 Ccbc 14341 Σcsu 15722 ∫citg 25653 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902 |
| This theorem is referenced by: lcmineqlem6 42035 |
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