lcmineqlem22 - Mathbox for metakunt

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Theorem lcmineqlem22 41222

Description: The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.)

**Hypotheses**
Ref	Expression
lcmineqlem22.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
lcmineqlem22.2	⊢ (𝜑 → 4 ≤ 𝑁)

**Assertion**
Ref	Expression
lcmineqlem22	⊢ (𝜑 → ((2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))) ∧ (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))

**Proof of Theorem lcmineqlem22**
Step	Hyp	Ref	Expression
1		2re 12291	. . . . 5 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
3		2nn0 12494	. . . . . . 7 ⊢ 2 ∈ ℕ₀
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ₀)
5		lcmineqlem22.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12537	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	4, 6	nn0mulcld 12542	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
8		1nn0 12493	. . . . . 6 ⊢ 1 ∈ ℕ₀
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
10	7, 9	nn0addcld 12541	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ₀)
11	2, 10	reexpcld 14133	. . 3 ⊢ (𝜑 → (2↑((2 · 𝑁) + 1)) ∈ ℝ)
12	7, 4	nn0addcld 12541	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 2) ∈ ℕ₀)
13	2, 12	reexpcld 14133	. . 3 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ∈ ℝ)
14		fz1ssnn 13537	. . . . . 6 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ
15		fzfi 13942	. . . . . 6 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin
16		lcmfnncl 16571	. . . . . 6 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
17	14, 15, 16	mp2an 689	. . . . 5 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ
18	17	a1i 11	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ)
19	18	nnred 12232	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ)
20		1red 11220	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
21	5	nnred 12232	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
22	2, 21	remulcld 11249	. . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
23		1lt2 12388	. . . . . . 7 ⊢ 1 < 2
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 < 2)
25	20, 2, 24	ltled 11367	. . . . 5 ⊢ (𝜑 → 1 ≤ 2)
26	20, 2, 22, 25	leadd2dd 11834	. . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ≤ ((2 · 𝑁) + 2))
27		2z 12599	. . . . . . . 8 ⊢ 2 ∈ ℤ
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
29	5	nnzd 12590	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
30	28, 29	zmulcld 12677	. . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℤ)
31	30	peano2zd 12674	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℤ)
32	30, 28	zaddcld 12675	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁) + 2) ∈ ℤ)
33	2, 31, 32, 24	leexp2d 14220	. . . 4 ⊢ (𝜑 → (((2 · 𝑁) + 1) ≤ ((2 · 𝑁) + 2) ↔ (2↑((2 · 𝑁) + 1)) ≤ (2↑((2 · 𝑁) + 2))))
34	26, 33	mpbid 231	. . 3 ⊢ (𝜑 → (2↑((2 · 𝑁) + 1)) ≤ (2↑((2 · 𝑁) + 2)))
35		lcmineqlem22.2	. . . 4 ⊢ (𝜑 → 4 ≤ 𝑁)
36	5, 35	lcmineqlem21 41221	. . 3 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
37	11, 13, 19, 34, 36	letrd 11376	. 2 ⊢ (𝜑 → (2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
38		fz1ssnn 13537	. . . . . 6 ⊢ (1...((2 · 𝑁) + 2)) ⊆ ℕ
39		fzfi 13942	. . . . . 6 ⊢ (1...((2 · 𝑁) + 2)) ∈ Fin
40		lcmfnncl 16571	. . . . . 6 ⊢ (((1...((2 · 𝑁) + 2)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 2)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℕ)
41	38, 39, 40	mp2an 689	. . . . 5 ⊢ (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℕ
42	41	a1i 11	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℕ)
43	42	nnred 12232	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℝ)
44	18	nnzd 12590	. . . . . . . 8 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℤ)
45	44, 32	jca 511	. . . . . . 7 ⊢ (𝜑 → ((lcm‘(1...((2 · 𝑁) + 1))) ∈ ℤ ∧ ((2 · 𝑁) + 2) ∈ ℤ))
46		dvdslcm 16540	. . . . . . 7 ⊢ (((lcm‘(1...((2 · 𝑁) + 1))) ∈ ℤ ∧ ((2 · 𝑁) + 2) ∈ ℤ) → ((lcm‘(1...((2 · 𝑁) + 1))) ∥ ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2)) ∧ ((2 · 𝑁) + 2) ∥ ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2))))
47	45, 46	syl 17	. . . . . 6 ⊢ (𝜑 → ((lcm‘(1...((2 · 𝑁) + 1))) ∥ ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2)) ∧ ((2 · 𝑁) + 2) ∥ ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2))))
48	47	simpld 494	. . . . 5 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∥ ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2)))
49		2nn 12290	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ)
51	50, 5	nnmulcld 12270	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ)
52	51, 50	nnaddcld 12269	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) + 2) ∈ ℕ)
53	52	lcmfunnnd 41184	. . . . . 6 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 2))) = ((lcm‘(1...(((2 · 𝑁) + 2) − 1))) lcm ((2 · 𝑁) + 2)))
54	22	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
55	2	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
56		1cnd 11214	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
57	54, 55, 56	addsubassd 11596	. . . . . . . . . 10 ⊢ (𝜑 → (((2 · 𝑁) + 2) − 1) = ((2 · 𝑁) + (2 − 1)))
58		2m1e1 12343	. . . . . . . . . . 11 ⊢ (2 − 1) = 1
59	58	oveq2i 7423	. . . . . . . . . 10 ⊢ ((2 · 𝑁) + (2 − 1)) = ((2 · 𝑁) + 1)
60	57, 59	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝑁) + 2) − 1) = ((2 · 𝑁) + 1))
61	60	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → (1...(((2 · 𝑁) + 2) − 1)) = (1...((2 · 𝑁) + 1)))
62	61	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → (lcm‘(1...(((2 · 𝑁) + 2) − 1))) = (lcm‘(1...((2 · 𝑁) + 1))))
63	62	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((lcm‘(1...(((2 · 𝑁) + 2) − 1))) lcm ((2 · 𝑁) + 2)) = ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2)))
64	53, 63	eqtrd 2771	. . . . 5 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 2))) = ((lcm‘(1...((2 · 𝑁) + 1))) lcm ((2 · 𝑁) + 2)))
65	48, 64	breqtrrd 5176	. . . 4 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 2))))
66	44, 42	jca 511	. . . . 5 ⊢ (𝜑 → ((lcm‘(1...((2 · 𝑁) + 1))) ∈ ℤ ∧ (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℕ))
67		dvdsle 16258	. . . . 5 ⊢ (((lcm‘(1...((2 · 𝑁) + 1))) ∈ ℤ ∧ (lcm‘(1...((2 · 𝑁) + 2))) ∈ ℕ) → ((lcm‘(1...((2 · 𝑁) + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 2))) → (lcm‘(1...((2 · 𝑁) + 1))) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))
68	66, 67	syl 17	. . . 4 ⊢ (𝜑 → ((lcm‘(1...((2 · 𝑁) + 1))) ∥ (lcm‘(1...((2 · 𝑁) + 2))) → (lcm‘(1...((2 · 𝑁) + 1))) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))
69	65, 68	mpd 15	. . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ≤ (lcm‘(1...((2 · 𝑁) + 2))))
70	13, 19, 43, 36, 69	letrd 11376	. 2 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2))))
71	37, 70	jca 511	1 ⊢ (𝜑 → ((2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))) ∧ (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Fincfn 8943 ℝcr 11113 1c1 11115 + caddc 11117 · cmul 11119 < clt 11253 ≤ cle 11254 − cmin 11449 ℕcn 12217 2c2 12272 4c4 12274 ℕ₀cn0 12477 ℤcz 12563 ...cfz 13489 ↑cexp 14032 ∥ cdvds 16202 lcm clcm 16530 lcmclcmf 16531

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-prod 15855 df-dvds 16203 df-gcd 16441 df-lcm 16532 df-lcmf 16533 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-ovol 25214 df-vol 25215 df-mbf 25369 df-itg1 25370 df-itg2 25371 df-ibl 25372 df-itg 25373 df-0p 25420 df-limc 25616 df-dv 25617

This theorem is referenced by: lcmineqlem23 41223

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