lcmineqlem20 - Mathbox for metakunt

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Theorem lcmineqlem20 42066

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 12260	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2nn0 12523	. . . . . 6 ⊢ 2 ∈ ℕ₀
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
5	1	nnnn0d 12567	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nn0mulcld 12572	. . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
7		2re 12319	. . . . 5 ⊢ 2 ∈ ℝ
8		reexpcl 14101	. . . . 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 11270	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ)
12	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
13	12, 2	remulcld 11270	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
14		1red 11241	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
15	13, 14	readdcld 11269	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℝ)
16		2nn 12318	. . . . . . . 8 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ)
18	17, 1	nnmulcld 12298	. . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
19	5	nn0ge0d 12570	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
20	17	nnge1d 12293	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 2)
21	2, 12, 19, 20	lemulge12d 12185	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
22	18, 5, 21	bccl2d 42009	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
23	22	nnred 12260	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ)
24	15, 23	remulcld 11270	. . 3 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∈ ℝ)
25	2, 24	remulcld 11270	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ∈ ℝ)
26		fz1ssnn 13577	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
27		fzfi 13995	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
28		lcmfnncl 16653	. . . . 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 12260	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ)
32	5	lcmineqlem17 42063	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
33	1	nnrpd 13054	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
34	10, 24, 33	lemul2d 13100	. . 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 11268	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
37	15	recnd 11268	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℂ)
38	23	recnd 11268	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ)
39	36, 37, 38	mulassd 11263	. . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) = (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
40	1	lcmineqlem19 42065	. . . 4 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
41	18	peano2nnd 12262	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
42	1, 41	nnmulcld 12298	. . . . . . 7 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁) + 1)) ∈ ℕ)
43	42, 22	nnmulcld 12298	. . . . . 6 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℕ)
44	43	nnzd 12620	. . . . 5 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ)
45		dvdsle 16334	. . . . 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 5148	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
49	11, 25, 31, 35, 48	letrd 11397	1 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2109 ⊆ wss 3931 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 Fincfn 8964 ℝcr 11133 1c1 11135 + caddc 11137 · cmul 11139 ≤ cle 11275 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 ℤcz 12593 ...cfz 13529 ↑cexp 14084 Ccbc 14325 ∥ cdvds 16277 lcmclcmf 16613

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-symdif 4233 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-prod 15925 df-dvds 16278 df-gcd 16519 df-lcm 16614 df-lcmf 16615 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25422 df-vol 25423 df-mbf 25577 df-itg1 25578 df-itg2 25579 df-ibl 25580 df-itg 25581 df-0p 25628 df-limc 25824 df-dv 25825

This theorem is referenced by: lcmineqlem21 42067

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