lcmineqlem19 - Mathbox for metakunt

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Theorem lcmineqlem19 42486

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

**Hypothesis**
Ref	Expression
lcmineqlem19.1	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
lcmineqlem19	⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))

**Proof of Theorem lcmineqlem19**
Step	Hyp	Ref	Expression
1		lcmineqlem19.1	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		2nn 12254	. . . . 5 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ)
4	3, 1	nnmulcld 12230	. . 3 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
5	4	peano2nnd 12191	. 2 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
6	1	nnnn0d 12498	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1	nnred 12189	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		2re 12255	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	6	nn0ge0d 12501	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
11	3	nnge1d 12225	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
12	7, 9, 10, 11	lemulge12d 12094	. . 3 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
13	4, 6, 12	bccl2d 42430	. 2 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
14		fz1ssnn 13509	. . . 4 ⊢ (1...(2 · 𝑁)) ⊆ ℕ
15		fzfi 13934	. . . 4 ⊢ (1...(2 · 𝑁)) ∈ Fin
16		lcmfnncl 16598	. . . 4 ⊢ (((1...(2 · 𝑁)) ⊆ ℕ ∧ (1...(2 · 𝑁)) ∈ Fin) → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
17	14, 15, 16	mp2an 693	. . 3 ⊢ (lcm‘(1...(2 · 𝑁))) ∈ ℕ
18	17	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
19		fz1ssnn 13509	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
20		fzfi 13934	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
21		lcmfnncl 16598	. . . 4 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
22	19, 20, 21	mp2an 693	. . 3 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
23	22	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
24	1, 4, 12	lcmineqlem16 42483	. 2 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...(2 · 𝑁))))
25	1	lcmineqlem18 42485	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
26	1	peano2nnd 12191	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27	9, 7	remulcld 11175	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
28		1red 11145	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
29	7, 27, 28, 12	leadd1dd 11764	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ≤ ((2 · 𝑁) + 1))
30	26, 5, 29	lcmineqlem16 42483	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
31	25, 30	eqbrtrrd 5109	. 2 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
32	18	nnzd 12550	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℤ)
33	5	nnzd 12550	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
34	32, 33	jca 511	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ))
35		dvdslcm 16567	. . . . 5 ⊢ (((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ) → ((lcm‘(1...(2 · 𝑁))) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)) ∧ ((2 · 𝑁) + 1) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1))))
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)) ∧ ((2 · 𝑁) + 1) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1))))
37	36	simpld 494	. . 3 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
38	5	lcmfunnnd 42451	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)))
39	27	recnd 11173	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
40		1cnd 11139	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
41	39, 40	pncand 11506	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) + 1) − 1) = (2 · 𝑁))
42	41	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → (1...(((2 · 𝑁) + 1) − 1)) = (1...(2 · 𝑁)))
43	42	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (lcm‘(1...(((2 · 𝑁) + 1) − 1))) = (lcm‘(1...(2 · 𝑁))))
44	43	oveq1d 7382	. . . 4 ⊢ (𝜑 → ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
45	38, 44	eqtrd 2771	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
46	37, 45	breqtrrd 5113	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
47	1	nnzd 12550	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
48		2z 12559	. . . . . 6 ⊢ 2 ∈ ℤ
49		1z 12557	. . . . . 6 ⊢ 1 ∈ ℤ
50		gcdaddm 16494	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
51	48, 49, 50	mp3an13 1455	. . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
52	47, 51	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
53	40, 39	addcomd 11348	. . . . 5 ⊢ (𝜑 → (1 + (2 · 𝑁)) = ((2 · 𝑁) + 1))
54	53	oveq2d 7383	. . . 4 ⊢ (𝜑 → (𝑁 gcd (1 + (2 · 𝑁))) = (𝑁 gcd ((2 · 𝑁) + 1)))
55	52, 54	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd ((2 · 𝑁) + 1)))
56		gcd1 16497	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = 1)
57	47, 56	syl 17	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = 1)
58	55, 57	eqtr3d 2773	. 2 ⊢ (𝜑 → (𝑁 gcd ((2 · 𝑁) + 1)) = 1)
59	1, 5, 13, 18, 23, 24, 31, 46, 58	lcmineqlem14 42481	1 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3889 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 Fincfn 8893 ℝcr 11037 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11377 ℕcn 12174 2c2 12236 ℤcz 12524 ...cfz 13461 Ccbc 14264 ∥ cdvds 16221 gcd cgcd 16463 lcm clcm 16557 lcmclcmf 16558

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-symdif 4193 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-prod 15869 df-dvds 16222 df-gcd 16464 df-lcm 16559 df-lcmf 16560 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637 df-limc 25833 df-dv 25834

This theorem is referenced by: lcmineqlem20 42487

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