lcmineqlem20 - Mathbox for metakunt

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Theorem lcmineqlem20 42050

Description: Inequality for lcm lemma. (Contributed by metakunt, 12-May-2024.)

**Hypothesis**
Ref	Expression
lcmineqlem20.1	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
lcmineqlem20	⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

**Proof of Theorem lcmineqlem20**
Step	Hyp	Ref	Expression
1		lcmineqlem20.1	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnred 12282	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2nn0 12545	. . . . . 6 ⊢ 2 ∈ ℕ₀
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
5	1	nnnn0d 12589	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nn0mulcld 12594	. . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
7		2re 12341	. . . . 5 ⊢ 2 ∈ ℝ
8		reexpcl 14120	. . . . 5 ⊢ ((2 ∈ ℝ ∧ (2 · 𝑁) ∈ ℕ₀) → (2↑(2 · 𝑁)) ∈ ℝ)
9	7, 8	mpan 690	. . . 4 ⊢ ((2 · 𝑁) ∈ ℕ₀ → (2↑(2 · 𝑁)) ∈ ℝ)
10	6, 9	syl 17	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ)
11	2, 10	remulcld 11292	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ)
12	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
13	12, 2	remulcld 11292	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
14		1red 11263	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
15	13, 14	readdcld 11291	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℝ)
16		2nn 12340	. . . . . . . 8 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ)
18	17, 1	nnmulcld 12320	. . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
19	5	nn0ge0d 12592	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
20	17	nnge1d 12315	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 2)
21	2, 12, 19, 20	lemulge12d 12207	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
22	18, 5, 21	bccl2d 41993	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
23	22	nnred 12282	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ)
24	15, 23	remulcld 11292	. . 3 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∈ ℝ)
25	2, 24	remulcld 11292	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ∈ ℝ)
26		fz1ssnn 13596	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
27		fzfi 14014	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
28		lcmfnncl 16667	. . . . 5 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
29	26, 27, 28	mp2an 692	. . . 4 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
30	29	a1i 11	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
31	30	nnred 12282	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ)
32	5	lcmineqlem17 42047	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
33	1	nnrpd 13076	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
34	10, 24, 33	lemul2d 13122	. . 3 ⊢ (𝜑 → ((2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ↔ (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))))
35	32, 34	mpbid 232	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
36	2	recnd 11290	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
37	15	recnd 11290	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℂ)
38	23	recnd 11290	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ)
39	36, 37, 38	mulassd 11285	. . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) = (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
40	1	lcmineqlem19 42049	. . . 4 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
41	18	peano2nnd 12284	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
42	1, 41	nnmulcld 12320	. . . . . . 7 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁) + 1)) ∈ ℕ)
43	42, 22	nnmulcld 12320	. . . . . 6 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℕ)
44	43	nnzd 12642	. . . . 5 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ)
45		dvdsle 16348	. . . . 5 ⊢ ((((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ ∧ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) → (((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))) → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))))
46	44, 30, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → (((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))) → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))))
47	40, 46	mpd 15	. . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
48	39, 47	eqbrtrrd 5166	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
49	11, 25, 31, 35, 48	letrd 11419	1 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℝcr 11155 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 ℕcn 12267 2c2 12322 ℕ₀cn0 12528 ℤcz 12615 ...cfz 13548 ↑cexp 14103 Ccbc 14342 ∥ cdvds 16291 lcmclcmf 16627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-symdif 4252 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-prod 15941 df-dvds 16292 df-gcd 16533 df-lcm 16628 df-lcmf 16629 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-ovol 25500 df-vol 25501 df-mbf 25655 df-itg1 25656 df-itg2 25657 df-ibl 25658 df-itg 25659 df-0p 25706 df-limc 25902 df-dv 25903

This theorem is referenced by: lcmineqlem21 42051

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