lcmineqlem19 - Mathbox for metakunt

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Theorem lcmineqlem19 42500

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 12245	. . . . 5 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ)
4	3, 1	nnmulcld 12221	. . 3 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
5	4	peano2nnd 12182	. 2 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
6	1	nnnn0d 12489	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1	nnred 12180	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		2re 12246	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	6	nn0ge0d 12492	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
11	3	nnge1d 12216	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
12	7, 9, 10, 11	lemulge12d 12085	. . 3 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
13	4, 6, 12	bccl2d 42444	. 2 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
14		fz1ssnn 13500	. . . 4 ⊢ (1...(2 · 𝑁)) ⊆ ℕ
15		fzfi 13925	. . . 4 ⊢ (1...(2 · 𝑁)) ∈ Fin
16		lcmfnncl 16589	. . . 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 13500	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
20		fzfi 13925	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
21		lcmfnncl 16589	. . . 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 42497	. 2 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...(2 · 𝑁))))
25	1	lcmineqlem18 42499	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
26	1	peano2nnd 12182	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27	9, 7	remulcld 11166	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
28		1red 11136	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
29	7, 27, 28, 12	leadd1dd 11755	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ≤ ((2 · 𝑁) + 1))
30	26, 5, 29	lcmineqlem16 42497	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
31	25, 30	eqbrtrrd 5110	. 2 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
32	18	nnzd 12541	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℤ)
33	5	nnzd 12541	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
34	32, 33	jca 511	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ))
35		dvdslcm 16558	. . . . 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 42465	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)))
39	27	recnd 11164	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
40		1cnd 11130	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
41	39, 40	pncand 11497	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) + 1) − 1) = (2 · 𝑁))
42	41	oveq2d 7376	. . . . . 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 7375	. . . 4 ⊢ (𝜑 → ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
45	38, 44	eqtrd 2772	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
46	37, 45	breqtrrd 5114	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
47	1	nnzd 12541	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
48		2z 12550	. . . . . 6 ⊢ 2 ∈ ℤ
49		1z 12548	. . . . . 6 ⊢ 1 ∈ ℤ
50		gcdaddm 16485	. . . . . 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 11339	. . . . 5 ⊢ (𝜑 → (1 + (2 · 𝑁)) = ((2 · 𝑁) + 1))
54	53	oveq2d 7376	. . . 4 ⊢ (𝜑 → (𝑁 gcd (1 + (2 · 𝑁))) = (𝑁 gcd ((2 · 𝑁) + 1)))
55	52, 54	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd ((2 · 𝑁) + 1)))
56		gcd1 16488	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = 1)
57	47, 56	syl 17	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = 1)
58	55, 57	eqtr3d 2774	. 2 ⊢ (𝜑 → (𝑁 gcd ((2 · 𝑁) + 1)) = 1)
59	1, 5, 13, 18, 23, 24, 31, 46, 58	lcmineqlem14 42495	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 3890 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 ℝcr 11028 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 ℕcn 12165 2c2 12227 ℤcz 12515 ...cfz 13452 Ccbc 14255 ∥ cdvds 16212 gcd cgcd 16454 lcm clcm 16548 lcmclcmf 16549

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 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-omul 8403 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-dvds 16213 df-gcd 16455 df-lcm 16550 df-lcmf 16551 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-ovol 25441 df-vol 25442 df-mbf 25596 df-itg1 25597 df-itg2 25598 df-ibl 25599 df-itg 25600 df-0p 25647 df-limc 25843 df-dv 25844

This theorem is referenced by: lcmineqlem20 42501

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