lcmineqlem19 - Mathbox for metakunt

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Theorem lcmineqlem19 42146

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 12204	. . . . 5 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ)
4	3, 1	nnmulcld 12184	. . 3 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
5	4	peano2nnd 12148	. 2 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
6	1	nnnn0d 12448	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1	nnred 12146	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		2re 12205	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	6	nn0ge0d 12451	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
11	3	nnge1d 12179	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
12	7, 9, 10, 11	lemulge12d 12066	. . 3 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
13	4, 6, 12	bccl2d 42090	. 2 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
14		fz1ssnn 13461	. . . 4 ⊢ (1...(2 · 𝑁)) ⊆ ℕ
15		fzfi 13885	. . . 4 ⊢ (1...(2 · 𝑁)) ∈ Fin
16		lcmfnncl 16546	. . . 4 ⊢ (((1...(2 · 𝑁)) ⊆ ℕ ∧ (1...(2 · 𝑁)) ∈ Fin) → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
17	14, 15, 16	mp2an 692	. . 3 ⊢ (lcm‘(1...(2 · 𝑁))) ∈ ℕ
18	17	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
19		fz1ssnn 13461	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
20		fzfi 13885	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
21		lcmfnncl 16546	. . . 4 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
22	19, 20, 21	mp2an 692	. . 3 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
23	22	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
24	1, 4, 12	lcmineqlem16 42143	. 2 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...(2 · 𝑁))))
25	1	lcmineqlem18 42145	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
26	1	peano2nnd 12148	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27	9, 7	remulcld 11148	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
28		1red 11119	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
29	7, 27, 28, 12	leadd1dd 11737	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ≤ ((2 · 𝑁) + 1))
30	26, 5, 29	lcmineqlem16 42143	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
31	25, 30	eqbrtrrd 5117	. 2 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
32	18	nnzd 12501	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℤ)
33	5	nnzd 12501	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
34	32, 33	jca 511	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ))
35		dvdslcm 16515	. . . . 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 42111	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)))
39	27	recnd 11146	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
40		1cnd 11113	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
41	39, 40	pncand 11479	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) + 1) − 1) = (2 · 𝑁))
42	41	oveq2d 7368	. . . . . 6 ⊢ (𝜑 → (1...(((2 · 𝑁) + 1) − 1)) = (1...(2 · 𝑁)))
43	42	fveq2d 6832	. . . . 5 ⊢ (𝜑 → (lcm‘(1...(((2 · 𝑁) + 1) − 1))) = (lcm‘(1...(2 · 𝑁))))
44	43	oveq1d 7367	. . . 4 ⊢ (𝜑 → ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
45	38, 44	eqtrd 2766	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
46	37, 45	breqtrrd 5121	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
47	1	nnzd 12501	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
48		2z 12510	. . . . . 6 ⊢ 2 ∈ ℤ
49		1z 12508	. . . . . 6 ⊢ 1 ∈ ℤ
50		gcdaddm 16442	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
51	48, 49, 50	mp3an13 1454	. . . . 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 11321	. . . . 5 ⊢ (𝜑 → (1 + (2 · 𝑁)) = ((2 · 𝑁) + 1))
54	53	oveq2d 7368	. . . 4 ⊢ (𝜑 → (𝑁 gcd (1 + (2 · 𝑁))) = (𝑁 gcd ((2 · 𝑁) + 1)))
55	52, 54	eqtrd 2766	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd ((2 · 𝑁) + 1)))
56		gcd1 16445	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = 1)
57	47, 56	syl 17	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = 1)
58	55, 57	eqtr3d 2768	. 2 ⊢ (𝜑 → (𝑁 gcd ((2 · 𝑁) + 1)) = 1)
59	1, 5, 13, 18, 23, 24, 31, 46, 58	lcmineqlem14 42141	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 1541 ∈ wcel 2111 ⊆ wss 3897 class class class wbr 5093 ‘cfv 6487 (class class class)co 7352 Fincfn 8875 ℝcr 11011 1c1 11013 + caddc 11015 · cmul 11017 − cmin 11350 ℕcn 12131 2c2 12186 ℤcz 12474 ...cfz 13413 Ccbc 14215 ∥ cdvds 16169 gcd cgcd 16411 lcm clcm 16505 lcmclcmf 16506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cc 10332 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-symdif 4202 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-disj 5061 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-omul 8396 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9800 df-card 9838 df-acn 9841 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ioc 13256 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-prod 15817 df-dvds 16170 df-gcd 16412 df-lcm 16507 df-lcmf 16508 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-mulg 18987 df-cntz 19235 df-cmn 19700 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-cmp 23308 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-ovol 25398 df-vol 25399 df-mbf 25553 df-itg1 25554 df-itg2 25555 df-ibl 25556 df-itg 25557 df-0p 25604 df-limc 25800 df-dv 25801

This theorem is referenced by: lcmineqlem20 42147

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