lcmineqlem19 - Mathbox for metakunt

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Theorem lcmineqlem19 42847

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 12325	. . . . 5 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ)
4	3, 1	nnmulcld 12300	. . 3 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
5	4	peano2nnd 12261	. 2 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ)
6	1	nnnn0d 12576	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1	nnred 12259	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		2re 12326	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	6	nn0ge0d 12579	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
11	3	nnge1d 12295	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
12	7, 9, 10, 11	lemulge12d 12164	. . 3 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁))
13	4, 6, 12	bccl2d 42791	. 2 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ)
14		fz1ssnn 13595	. . . 4 ⊢ (1...(2 · 𝑁)) ⊆ ℕ
15		fzfi 14021	. . . 4 ⊢ (1...(2 · 𝑁)) ∈ Fin
16		lcmfnncl 16704	. . . 4 ⊢ (((1...(2 · 𝑁)) ⊆ ℕ ∧ (1...(2 · 𝑁)) ∈ Fin) → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
17	14, 15, 16	mp2an 705	. . 3 ⊢ (lcm‘(1...(2 · 𝑁))) ∈ ℕ
18	17	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℕ)
19		fz1ssnn 13595	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
20		fzfi 14021	. . . 4 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
21		lcmfnncl 16704	. . . 4 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
22	19, 20, 21	mp2an 705	. . 3 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
23	22	a1i 11	. 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
24	1, 4, 12	lcmineqlem16 42844	. 2 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...(2 · 𝑁))))
25	1	lcmineqlem18 42846	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
26	1	peano2nnd 12261	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27	9, 7	remulcld 11250	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
28		1red 11220	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
29	7, 27, 28, 12	leadd1dd 11839	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ≤ ((2 · 𝑁) + 1))
30	26, 5, 29	lcmineqlem16 42844	. . 3 ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
31	25, 30	eqbrtrrd 5137	. 2 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
32	18	nnzd 12628	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∈ ℤ)
33	5	nnzd 12628	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
34	32, 33	jca 521	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...(2 · 𝑁))) ∈ ℤ ∧ ((2 · 𝑁) + 1) ∈ ℤ))
35		dvdslcm 16673	. . . . 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 18	. . . 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 500	. . 3 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
38	5	lcmfunnnd 42812	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)))
39	27	recnd 11248	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
40		1cnd 11213	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
41	39, 40	pncand 11581	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) + 1) − 1) = (2 · 𝑁))
42	41	oveq2d 7432	. . . . . 6 ⊢ (𝜑 → (1...(((2 · 𝑁) + 1) − 1)) = (1...(2 · 𝑁)))
43	42	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (lcm‘(1...(((2 · 𝑁) + 1) − 1))) = (lcm‘(1...(2 · 𝑁))))
44	43	oveq1d 7431	. . . 4 ⊢ (𝜑 → ((lcm‘(1...(((2 · 𝑁) + 1) − 1))) lcm ((2 · 𝑁) + 1)) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
45	38, 44	eqtrd 2800	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) = ((lcm‘(1...(2 · 𝑁))) lcm ((2 · 𝑁) + 1)))
46	37, 45	breqtrrd 5141	. 2 ⊢ (𝜑 → (lcm‘(1...(2 · 𝑁))) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
47	1	nnzd 12628	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
48		2z 12637	. . . . . 6 ⊢ 2 ∈ ℤ
49		1z 12635	. . . . . 6 ⊢ 1 ∈ ℤ
50		gcdaddm 16600	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
51	48, 49, 50	mp3an13 1481	. . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
52	47, 51	syl 18	. . . 4 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd (1 + (2 · 𝑁))))
53	40, 39	addcomd 11423	. . . . 5 ⊢ (𝜑 → (1 + (2 · 𝑁)) = ((2 · 𝑁) + 1))
54	53	oveq2d 7432	. . . 4 ⊢ (𝜑 → (𝑁 gcd (1 + (2 · 𝑁))) = (𝑁 gcd ((2 · 𝑁) + 1)))
55	52, 54	eqtrd 2800	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = (𝑁 gcd ((2 · 𝑁) + 1)))
56		gcd1 16603	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = 1)
57	47, 56	syl 18	. . 3 ⊢ (𝜑 → (𝑁 gcd 1) = 1)
58	55, 57	eqtr3d 2802	. 2 ⊢ (𝜑 → (𝑁 gcd ((2 · 𝑁) + 1)) = 1)
59	1, 5, 13, 18, 23, 24, 31, 46, 58	lcmineqlem14 42842	1 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 class class class wbr 5111 ‘cfv 6540 (class class class)co 7416 Fincfn 8945 ℝcr 11110 1c1 11112 + caddc 11114 · cmul 11116 − cmin 11452 ℕcn 12244 2c2 12306 ℤcz 12602 ...cfz 13546 Ccbc 14351 ∥ cdvds 16327 gcd cgcd 16569 lcm clcm 16663 lcmclcmf 16664

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cc 10430 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4206 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-ofr 7681 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-dju 9899 df-card 9937 df-acn 9940 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-q 12984 df-rp 13028 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13547 df-fzo 13695 df-fl 13838 df-mod 13916 df-seq 14051 df-exp 14111 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-limsup 15541 df-clim 15558 df-rlim 15559 df-sum 15757 df-prod 15976 df-dvds 16328 df-gcd 16570 df-lcm 16665 df-lcmf 16666 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-rest 17492 df-topn 17493 df-0g 17511 df-gsum 17512 df-topgen 17513 df-pt 17514 df-prds 17517 df-xrs 17573 df-qtop 17578 df-imas 17579 df-xps 17581 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-mulg 19157 df-cntz 19410 df-cmn 19875 df-psmet 21543 df-xmet 21544 df-met 21545 df-bl 21546 df-mopn 21547 df-fbas 21548 df-fg 21549 df-cnfld 21552 df-top 23080 df-topon 23097 df-topsp 23119 df-bases 23132 df-cld 23205 df-ntr 23206 df-cls 23207 df-nei 23284 df-lp 23322 df-perf 23323 df-cn 23413 df-cnp 23414 df-haus 23501 df-cmp 23573 df-tx 23748 df-hmeo 23941 df-fil 24032 df-fm 24124 df-flim 24125 df-flf 24126 df-xms 24506 df-ms 24507 df-tms 24508 df-cncf 25066 df-ovol 25652 df-vol 25653 df-mbf 25807 df-itg1 25808 df-itg2 25809 df-ibl 25810 df-itg 25811 df-0p 25858 df-limc 26054 df-dv 26055

This theorem is used by: lcmineqlem20 42848

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