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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem15 | Structured version Visualization version GIF version |
Description: F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.) |
Ref | Expression |
---|---|
lcmineqlem15.1 | ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 |
lcmineqlem15.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
lcmineqlem15.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
lcmineqlem15.4 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
Ref | Expression |
---|---|
lcmineqlem15 | ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmineqlem15.1 | . . 3 ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 | |
2 | lcmineqlem15.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | lcmineqlem15.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
4 | lcmineqlem15.4 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | |
5 | 1, 2, 3, 4 | lcmineqlem6 41358 | . 2 ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ) |
6 | fz1ssnn 13528 | . . . . . 6 ⊢ (1...𝑁) ⊆ ℕ | |
7 | fzfi 13933 | . . . . . 6 ⊢ (1...𝑁) ∈ Fin | |
8 | lcmfnncl 16562 | . . . . . 6 ⊢ (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ) | |
9 | 6, 7, 8 | mp2an 689 | . . . . 5 ⊢ (lcm‘(1...𝑁)) ∈ ℕ |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (lcm‘(1...𝑁)) ∈ ℕ) |
11 | 10 | nnred 12223 | . . 3 ⊢ (𝜑 → (lcm‘(1...𝑁)) ∈ ℝ) |
12 | 1, 3, 2, 4 | lcmineqlem13 41365 | . . . 4 ⊢ (𝜑 → 𝐹 = (1 / (𝑀 · (𝑁C𝑀)))) |
13 | 1red 11211 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
14 | 3 | nnnn0d 12528 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
15 | 2, 14, 4 | bccl2d 41316 | . . . . . . 7 ⊢ (𝜑 → (𝑁C𝑀) ∈ ℕ) |
16 | 3, 15 | nnmulcld 12261 | . . . . . 6 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∈ ℕ) |
17 | 16 | nnred 12223 | . . . . 5 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∈ ℝ) |
18 | 16 | nnne0d 12258 | . . . . 5 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ≠ 0) |
19 | 13, 17, 18 | redivcld 12038 | . . . 4 ⊢ (𝜑 → (1 / (𝑀 · (𝑁C𝑀))) ∈ ℝ) |
20 | 12, 19 | eqeltrd 2825 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ℝ) |
21 | 10 | nngt0d 12257 | . . 3 ⊢ (𝜑 → 0 < (lcm‘(1...𝑁))) |
22 | nnrecgt0 12251 | . . . . 5 ⊢ ((𝑀 · (𝑁C𝑀)) ∈ ℕ → 0 < (1 / (𝑀 · (𝑁C𝑀)))) | |
23 | 16, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 0 < (1 / (𝑀 · (𝑁C𝑀)))) |
24 | 23, 12 | breqtrrd 5166 | . . 3 ⊢ (𝜑 → 0 < 𝐹) |
25 | 11, 20, 21, 24 | mulgt0d 11365 | . 2 ⊢ (𝜑 → 0 < ((lcm‘(1...𝑁)) · 𝐹)) |
26 | elnnz 12564 | . 2 ⊢ (((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ ↔ (((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ ∧ 0 < ((lcm‘(1...𝑁)) · 𝐹))) | |
27 | 5, 25, 26 | sylanbrc 582 | 1 ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Fincfn 8934 ℝcr 11104 0cc0 11105 1c1 11106 · cmul 11110 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℤcz 12554 [,]cicc 13323 ...cfz 13480 ↑cexp 14023 Ccbc 14258 lcmclcmf 16522 ∫citg 25457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-symdif 4234 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-prod 15846 df-dvds 16194 df-lcmf 16524 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-lp 22950 df-perf 22951 df-cn 23041 df-cnp 23042 df-haus 23129 df-cmp 23201 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cncf 24708 df-ovol 25303 df-vol 25304 df-mbf 25458 df-itg1 25459 df-itg2 25460 df-ibl 25461 df-itg 25462 df-0p 25509 df-limc 25705 df-dv 25706 |
This theorem is referenced by: lcmineqlem16 41368 |
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