lcmineqlem15 - Mathbox for metakunt

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Theorem lcmineqlem15 40896

Description: F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
lcmineqlem15	⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ)

Distinct variable groups: 𝑥,𝑀 𝑥,𝑁 𝜑,𝑥 Allowed substitution hint: 𝐹(𝑥)

**Proof of Theorem lcmineqlem15**
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 40887	. 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 690	. . . . 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 40894	. . . 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 40845	. . . . . . 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 2833	. . 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 5175	. . 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 583	1 ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℤcz 12554 [,]cicc 13323 ...cfz 13480 ↑cexp 14023 Ccbc 14258 lcmclcmf 16522 ∫citg 25126

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 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 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-itg 25131 df-0p 25178 df-limc 25374 df-dv 25375

This theorem is referenced by: lcmineqlem16 40897

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