lcmineqlem19 - Mathbox for metakunt

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Theorem lcmineqlem19 42297

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 12218	. . . . 5 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ)
4	3, 1	nnmulcld 12198	. . 3 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
5	4	peano2nnd 12162	. 2 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
6	1	nnnn0d 12462	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1	nnred 12160	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		2re 12219	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	6	nn0ge0d 12465	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
11	3	nnge1d 12193	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
12	7, 9, 10, 11	lemulge12d 12080	. . 3 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
13	4, 6, 12	bccl2d 42241	. 2 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
14		fz1ssnn 13471	. . . 4 ⊢ (1...(2 · 𝑁)) ⊆ ℕ
15		fzfi 13895	. . . 4 ⊢ (1...(2 · 𝑁)) ∈ Fin
16		lcmfnncl 16556	. . . 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 13471	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
20		fzfi 13895	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
21		lcmfnncl 16556	. . . 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 42294	. 2 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...(2 · 𝑁))))
25	1	lcmineqlem18 42296	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
26	1	peano2nnd 12162	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27	9, 7	remulcld 11162	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
28		1red 11133	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
29	7, 27, 28, 12	leadd1dd 11751	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ≤ ((2 · 𝑁) + 1))
30	26, 5, 29	lcmineqlem16 42294	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
31	25, 30	eqbrtrrd 5122	. 2 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
32	18	nnzd 12514	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℤ)
33	5	nnzd 12514	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
34	32, 33	jca 511	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ))
35		dvdslcm 16525	. . . . 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 42262	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)))
39	27	recnd 11160	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
40		1cnd 11127	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
41	39, 40	pncand 11493	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) + 1) − 1) = (2 · 𝑁))
42	41	oveq2d 7374	. . . . . 6 ⊢ (𝜑 → (1...(((2 · 𝑁) + 1) − 1)) = (1...(2 · 𝑁)))
43	42	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (lcm‘(1...(((2 · 𝑁) + 1) − 1))) = (lcm‘(1...(2 · 𝑁))))
44	43	oveq1d 7373	. . . 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 5126	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
47	1	nnzd 12514	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
48		2z 12523	. . . . . 6 ⊢ 2 ∈ ℤ
49		1z 12521	. . . . . 6 ⊢ 1 ∈ ℤ
50		gcdaddm 16452	. . . . . 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 11335	. . . . 5 ⊢ (𝜑 → (1 + (2 · 𝑁)) = ((2 · 𝑁) + 1))
54	53	oveq2d 7374	. . . 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 16455	. . . 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 42292	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 2113 ⊆ wss 3901 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ℝcr 11025 1c1 11027 + caddc 11029 · cmul 11031 − cmin 11364 ℕcn 12145 2c2 12200 ℤcz 12488 ...cfz 13423 Ccbc 14225 ∥ cdvds 16179 gcd cgcd 16421 lcm clcm 16515 lcmclcmf 16516

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

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 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-symdif 4205 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-prod 15827 df-dvds 16180 df-gcd 16422 df-lcm 16517 df-lcmf 16518 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580 df-0p 25627 df-limc 25823 df-dv 25824

This theorem is referenced by: lcmineqlem20 42298

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