lcmineqlem20 - Mathbox for metakunt

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Theorem lcmineqlem20 39747

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 11828	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2nn0 12090	. . . . . 6 ⊢ 2 ∈ ℕ₀
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
5	1	nnnn0d 12133	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nn0mulcld 12138	. . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
7		2re 11887	. . . . 5 ⊢ 2 ∈ ℝ
8		reexpcl 13635	. . . . 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 10846	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ)
12	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
13	12, 2	remulcld 10846	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
14		1red 10817	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
15	13, 14	readdcld 10845	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℝ)
16		2nn 11886	. . . . . . . 8 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ)
18	17, 1	nnmulcld 11866	. . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
19	5	nn0ge0d 12136	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
20	17	nnge1d 11861	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 2)
21	2, 12, 19, 20	lemulge12d 11753	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
22	18, 5, 21	bccl2d 39691	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
23	22	nnred 11828	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ)
24	15, 23	remulcld 10846	. . 3 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∈ ℝ)
25	2, 24	remulcld 10846	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ∈ ℝ)
26		fz1ssnn 13126	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
27		fzfi 13528	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
28		lcmfnncl 16167	. . . . 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 11828	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ)
32	5	lcmineqlem17 39744	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
33	1	nnrpd 12609	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
34	10, 24, 33	lemul2d 12655	. . 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 235	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
36	2	recnd 10844	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
37	15	recnd 10844	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℂ)
38	23	recnd 10844	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ)
39	36, 37, 38	mulassd 10839	. . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) = (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
40	1	lcmineqlem19 39746	. . . 4 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
41	18	peano2nnd 11830	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
42	1, 41	nnmulcld 11866	. . . . . . 7 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁) + 1)) ∈ ℕ)
43	42, 22	nnmulcld 11866	. . . . . 6 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℕ)
44	43	nnzd 12264	. . . . 5 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ)
45		dvdsle 15852	. . . . 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 587	. . . 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 5067	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
49	11, 25, 31, 35, 48	letrd 10972	1 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 ℝcr 10711 1c1 10713 + caddc 10715 · cmul 10717 ≤ cle 10851 ℕcn 11813 2c2 11868 ℕ₀cn0 12073 ℤcz 12159 ...cfz 13078 ↑cexp 13618 Ccbc 13851 ∥ cdvds 15796 lcmclcmf 16127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-symdif 4147 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-disj 5009 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-prod 15449 df-dvds 15797 df-gcd 16035 df-lcm 16128 df-lcmf 16129 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-cmp 22256 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-ovol 24333 df-vol 24334 df-mbf 24488 df-itg1 24489 df-itg2 24490 df-ibl 24491 df-itg 24492 df-0p 24539 df-limc 24735 df-dv 24736

This theorem is referenced by: lcmineqlem21 39748

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