lcmineqlem20 - Mathbox for metakunt

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Theorem lcmineqlem20 42625

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 12218	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2nn0 12491	. . . . . 6 ⊢ 2 ∈ ℕ₀
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
5	1	nnnn0d 12535	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nn0mulcld 12540	. . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
7		2re 12285	. . . . 5 ⊢ 2 ∈ ℝ
8		reexpcl 14084	. . . . 5 ⊢ ((2 ∈ ℝ ∧ (2 · 𝑁) ∈ ℕ₀) → (2↑(2 · 𝑁)) ∈ ℝ)
9	7, 8	mpan 700	. . . 4 ⊢ ((2 · 𝑁) ∈ ℕ₀ → (2↑(2 · 𝑁)) ∈ ℝ)
10	6, 9	syl 17	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ)
11	2, 10	remulcld 11205	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ)
12	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
13	12, 2	remulcld 11205	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
14		1red 11175	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
15	13, 14	readdcld 11204	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℝ)
16		2nn 12284	. . . . . . . 8 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ)
18	17, 1	nnmulcld 12259	. . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
19	5	nn0ge0d 12538	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
20	17	nnge1d 12254	. . . . . . 7 ⊢ (𝜑 → 1 ≤ 2)
21	2, 12, 19, 20	lemulge12d 12123	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
22	18, 5, 21	bccl2d 42568	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
23	22	nnred 12218	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ)
24	15, 23	remulcld 11205	. . 3 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∈ ℝ)
25	2, 24	remulcld 11205	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ∈ ℝ)
26		fz1ssnn 13553	. . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
27		fzfi 13978	. . . . 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 702	. . . 4 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
30	29	a1i 11	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
31	30	nnred 12218	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ)
32	5	lcmineqlem17 42622	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
33	1	nnrpd 13028	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
34	10, 24, 33	lemul2d 13074	. . 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 234	. 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
36	2	recnd 11203	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
37	15	recnd 11203	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℂ)
38	23	recnd 11203	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ)
39	36, 37, 38	mulassd 11198	. . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) = (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))
40	1	lcmineqlem19 42624	. . . 4 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
41	18	peano2nnd 12220	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
42	1, 41	nnmulcld 12259	. . . . . . 7 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁) + 1)) ∈ ℕ)
43	42, 22	nnmulcld 12259	. . . . . 6 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℕ)
44	43	nnzd 12587	. . . . 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 593	. . . 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 5121	. 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
49	11, 25, 31, 35, 48	letrd 11333	1 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2141 ⊆ wss 3902 class class class wbr 5097 ‘cfv 6515 (class class class)co 7390 Fincfn 8920 ℝcr 11065 1c1 11067 + caddc 11069 · cmul 11071 ≤ cle 11210 ℕcn 12203 2c2 12265 ℕ₀cn0 12474 ℤcz 12561 ...cfz 13505 ↑cexp 14067 Ccbc 14308 ∥ cdvds 16276 lcmclcmf 16613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cc 10385 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4203 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5065 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-ofr 7655 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-oadd 8434 df-omul 8435 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-fi 9350 df-sup 9381 df-inf 9382 df-oi 9451 df-dju 9852 df-card 9890 df-acn 9893 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13506 df-fzo 13653 df-fl 13795 df-mod 13873 df-seq 14008 df-exp 14068 df-fac 14280 df-bc 14309 df-hash 14337 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15488 df-clim 15505 df-rlim 15506 df-sum 15704 df-prod 15924 df-dvds 16277 df-gcd 16519 df-lcm 16614 df-lcmf 16615 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 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 17522 df-qtop 17527 df-imas 17528 df-xps 17530 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-submnd 18808 df-mulg 19100 df-cntz 19347 df-cmn 19812 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-fbas 21408 df-fg 21409 df-cnfld 21412 df-top 22941 df-topon 22958 df-topsp 22980 df-bases 22993 df-cld 23066 df-ntr 23067 df-cls 23068 df-nei 23145 df-lp 23183 df-perf 23184 df-cn 23274 df-cnp 23275 df-haus 23362 df-cmp 23434 df-tx 23609 df-hmeo 23802 df-fil 23893 df-fm 23985 df-flim 23986 df-flf 23987 df-xms 24367 df-ms 24368 df-tms 24369 df-cncf 24927 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-limc 25915 df-dv 25916

This theorem is referenced by: lcmineqlem21 42626

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