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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 42035 | . 2 ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ) | 
| 6 | fz1ssnn 13595 | . . . . . 6 ⊢ (1...𝑁) ⊆ ℕ | |
| 7 | fzfi 14013 | . . . . . 6 ⊢ (1...𝑁) ∈ Fin | |
| 8 | lcmfnncl 16666 | . . . . . 6 ⊢ (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ) | |
| 9 | 6, 7, 8 | mp2an 692 | . . . . 5 ⊢ (lcm‘(1...𝑁)) ∈ ℕ | 
| 10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (lcm‘(1...𝑁)) ∈ ℕ) | 
| 11 | 10 | nnred 12281 | . . 3 ⊢ (𝜑 → (lcm‘(1...𝑁)) ∈ ℝ) | 
| 12 | 1, 3, 2, 4 | lcmineqlem13 42042 | . . . 4 ⊢ (𝜑 → 𝐹 = (1 / (𝑀 · (𝑁C𝑀)))) | 
| 13 | 1red 11262 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 14 | 3 | nnnn0d 12587 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | 
| 15 | 2, 14, 4 | bccl2d 41992 | . . . . . . 7 ⊢ (𝜑 → (𝑁C𝑀) ∈ ℕ) | 
| 16 | 3, 15 | nnmulcld 12319 | . . . . . 6 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∈ ℕ) | 
| 17 | 16 | nnred 12281 | . . . . 5 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∈ ℝ) | 
| 18 | 16 | nnne0d 12316 | . . . . 5 ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ≠ 0) | 
| 19 | 13, 17, 18 | redivcld 12095 | . . . 4 ⊢ (𝜑 → (1 / (𝑀 · (𝑁C𝑀))) ∈ ℝ) | 
| 20 | 12, 19 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ℝ) | 
| 21 | 10 | nngt0d 12315 | . . 3 ⊢ (𝜑 → 0 < (lcm‘(1...𝑁))) | 
| 22 | nnrecgt0 12309 | . . . . 5 ⊢ ((𝑀 · (𝑁C𝑀)) ∈ ℕ → 0 < (1 / (𝑀 · (𝑁C𝑀)))) | |
| 23 | 16, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 0 < (1 / (𝑀 · (𝑁C𝑀)))) | 
| 24 | 23, 12 | breqtrrd 5171 | . . 3 ⊢ (𝜑 → 0 < 𝐹) | 
| 25 | 11, 20, 21, 24 | mulgt0d 11416 | . 2 ⊢ (𝜑 → 0 < ((lcm‘(1...𝑁)) · 𝐹)) | 
| 26 | elnnz 12623 | . 2 ⊢ (((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ ↔ (((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ ∧ 0 < ((lcm‘(1...𝑁)) · 𝐹))) | |
| 27 | 5, 25, 26 | sylanbrc 583 | 1 ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℤcz 12613 [,]cicc 13390 ...cfz 13547 ↑cexp 14102 Ccbc 14341 lcmclcmf 16626 ∫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-prod 15940 df-dvds 16291 df-lcmf 16628 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: lcmineqlem16 42045 | 
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