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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem20 | Structured version Visualization version GIF version |
Description: Inequality for lcm lemma. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
lcmineqlem20.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
lcmineqlem20 | ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmineqlem20.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
2 | 1 | nnred 11828 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | 2nn0 12090 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℕ0) |
5 | 1 | nnnn0d 12133 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 4, 5 | nn0mulcld 12138 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ0) |
7 | 2re 11887 | . . . . 5 ⊢ 2 ∈ ℝ | |
8 | reexpcl 13635 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ (2 · 𝑁) ∈ ℕ0) → (2↑(2 · 𝑁)) ∈ ℝ) | |
9 | 7, 8 | mpan 690 | . . . 4 ⊢ ((2 · 𝑁) ∈ ℕ0 → (2↑(2 · 𝑁)) ∈ ℝ) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ) |
11 | 2, 10 | remulcld 10846 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ) |
12 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
13 | 12, 2 | remulcld 10846 | . . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
14 | 1red 10817 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
15 | 13, 14 | readdcld 10845 | . . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℝ) |
16 | 2nn 11886 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ) |
18 | 17, 1 | nnmulcld 11866 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ) |
19 | 5 | nn0ge0d 12136 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁) |
20 | 17 | nnge1d 11861 | . . . . . . 7 ⊢ (𝜑 → 1 ≤ 2) |
21 | 2, 12, 19, 20 | lemulge12d 11753 | . . . . . 6 ⊢ (𝜑 → 𝑁 ≤ (2 · 𝑁)) |
22 | 18, 5, 21 | bccl2d 39691 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ) |
23 | 22 | nnred 11828 | . . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ) |
24 | 15, 23 | remulcld 10846 | . . 3 ⊢ (𝜑 → (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ∈ ℝ) |
25 | 2, 24 | remulcld 10846 | . 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ∈ ℝ) |
26 | fz1ssnn 13126 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ | |
27 | fzfi 13528 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin | |
28 | lcmfnncl 16167 | . . . . 5 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) | |
29 | 26, 27, 28 | mp2an 692 | . . . 4 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) |
31 | 30 | nnred 11828 | . 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ) |
32 | 5 | lcmineqlem17 39744 | . . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) |
33 | 1 | nnrpd 12609 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ+) |
34 | 10, 24, 33 | lemul2d 12655 | . . 3 ⊢ (𝜑 → ((2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)) ↔ (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))))) |
35 | 32, 34 | mpbid 235 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))) |
36 | 2 | recnd 10844 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
37 | 15 | recnd 10844 | . . . 4 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℂ) |
38 | 23 | recnd 10844 | . . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ) |
39 | 36, 37, 38 | mulassd 10839 | . . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) = (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))) |
40 | 1 | lcmineqlem19 39746 | . . . 4 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1)))) |
41 | 18 | peano2nnd 11830 | . . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) + 1) ∈ ℕ) |
42 | 1, 41 | nnmulcld 11866 | . . . . . . 7 ⊢ (𝜑 → (𝑁 · ((2 · 𝑁) + 1)) ∈ ℕ) |
43 | 42, 22 | nnmulcld 11866 | . . . . . 6 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℕ) |
44 | 43 | nnzd 12264 | . . . . 5 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ) |
45 | dvdsle 15852 | . . . . 5 ⊢ ((((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∈ ℤ ∧ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) → (((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))) → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))) | |
46 | 44, 30, 45 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))) → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))) |
47 | 40, 46 | mpd 15 | . . 3 ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
48 | 39, 47 | eqbrtrrd 5067 | . 2 ⊢ (𝜑 → (𝑁 · (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
49 | 11, 25, 31, 35, 48 | letrd 10972 | 1 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 ℝcr 10711 1c1 10713 + caddc 10715 · cmul 10717 ≤ cle 10851 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ℤcz 12159 ...cfz 13078 ↑cexp 13618 Ccbc 13851 ∥ cdvds 15796 lcmclcmf 16127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-symdif 4147 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-disj 5009 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-prod 15449 df-dvds 15797 df-gcd 16035 df-lcm 16128 df-lcmf 16129 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-cmp 22256 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-ovol 24333 df-vol 24334 df-mbf 24488 df-itg1 24489 df-itg2 24490 df-ibl 24491 df-itg 24492 df-0p 24539 df-limc 24735 df-dv 24736 |
This theorem is referenced by: lcmineqlem21 39748 |
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