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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem21 | Structured version Visualization version GIF version |
Description: The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
lcmineqlem21.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
lcmineqlem21.2 | ⊢ (𝜑 → 4 ≤ 𝑁) |
Ref | Expression |
---|---|
lcmineqlem21 | ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12107 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℕ0) |
3 | 2 | nn0red 12151 | . . 3 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | lcmineqlem21.1 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | 4 | nnnn0d 12150 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 2, 5 | nn0mulcld 12155 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ0) |
7 | 6, 2 | nn0addcld 12154 | . . 3 ⊢ (𝜑 → ((2 · 𝑁) + 2) ∈ ℕ0) |
8 | 3, 7 | reexpcld 13733 | . 2 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ∈ ℝ) |
9 | 4 | nnred 11845 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
10 | 2rp 12591 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ+) |
12 | 2z 12209 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
14 | 4 | nnzd 12281 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
15 | 13, 14 | zmulcld 12288 | . . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℤ) |
16 | 11, 15 | rpexpcld 13814 | . . . 4 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ+) |
17 | 16 | rpred 12628 | . . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ) |
18 | 9, 17 | remulcld 10863 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ) |
19 | fz1ssnn 13143 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ | |
20 | fzfi 13545 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin | |
21 | lcmfnncl 16186 | . . . . 5 ⊢ (((1...((2 · 𝑁) + 1)) ⊆ ℕ ∧ (1...((2 · 𝑁) + 1)) ∈ Fin) → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) | |
22 | 19, 20, 21 | mp2an 692 | . . . 4 ⊢ (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ |
23 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℕ) |
24 | 23 | nnred 11845 | . 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ) |
25 | lcmineqlem21.2 | . . . 4 ⊢ (𝜑 → 4 ≤ 𝑁) | |
26 | 4re 11914 | . . . . . 6 ⊢ 4 ∈ ℝ | |
27 | 26 | a1i 11 | . . . . 5 ⊢ (𝜑 → 4 ∈ ℝ) |
28 | 27, 9, 16 | lemul1d 12671 | . . . 4 ⊢ (𝜑 → (4 ≤ 𝑁 ↔ (4 · (2↑(2 · 𝑁))) ≤ (𝑁 · (2↑(2 · 𝑁))))) |
29 | 25, 28 | mpbid 235 | . . 3 ⊢ (𝜑 → (4 · (2↑(2 · 𝑁))) ≤ (𝑁 · (2↑(2 · 𝑁)))) |
30 | 2cnd 11908 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
31 | 30, 2, 6 | expaddd 13718 | . . . . . 6 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = ((2↑(2 · 𝑁)) · (2↑2))) |
32 | sq2 13766 | . . . . . . 7 ⊢ (2↑2) = 4 | |
33 | 32 | oveq2i 7224 | . . . . . 6 ⊢ ((2↑(2 · 𝑁)) · (2↑2)) = ((2↑(2 · 𝑁)) · 4) |
34 | 31, 33 | eqtrdi 2794 | . . . . 5 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = ((2↑(2 · 𝑁)) · 4)) |
35 | 16 | rpcnd 12630 | . . . . . 6 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℂ) |
36 | 27 | recnd 10861 | . . . . . 6 ⊢ (𝜑 → 4 ∈ ℂ) |
37 | 35, 36 | mulcomd 10854 | . . . . 5 ⊢ (𝜑 → ((2↑(2 · 𝑁)) · 4) = (4 · (2↑(2 · 𝑁)))) |
38 | 34, 37 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = (4 · (2↑(2 · 𝑁)))) |
39 | 38 | breq1d 5063 | . . 3 ⊢ (𝜑 → ((2↑((2 · 𝑁) + 2)) ≤ (𝑁 · (2↑(2 · 𝑁))) ↔ (4 · (2↑(2 · 𝑁))) ≤ (𝑁 · (2↑(2 · 𝑁))))) |
40 | 29, 39 | mpbird 260 | . 2 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (𝑁 · (2↑(2 · 𝑁)))) |
41 | 4 | lcmineqlem20 39790 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
42 | 8, 18, 24, 40, 41 | letrd 10989 | 1 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 ℝcr 10728 1c1 10730 + caddc 10732 · cmul 10734 ≤ cle 10868 ℕcn 11830 2c2 11885 4c4 11887 ℕ0cn0 12090 ℤcz 12176 ℝ+crp 12586 ...cfz 13095 ↑cexp 13635 lcmclcmf 16146 |
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 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-symdif 4157 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-prod 15468 df-dvds 15816 df-gcd 16054 df-lcm 16147 df-lcmf 16148 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-cmp 22284 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-ovol 24361 df-vol 24362 df-mbf 24516 df-itg1 24517 df-itg2 24518 df-ibl 24519 df-itg 24520 df-0p 24567 df-limc 24763 df-dv 24764 |
This theorem is referenced by: lcmineqlem22 39792 |
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