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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 11936 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℕ0) |
3 | 2 | nn0red 11980 | . . 3 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | lcmineqlem21.1 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | 4 | nnnn0d 11979 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 2, 5 | nn0mulcld 11984 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ0) |
7 | 6, 2 | nn0addcld 11983 | . . 3 ⊢ (𝜑 → ((2 · 𝑁) + 2) ∈ ℕ0) |
8 | 3, 7 | reexpcld 13562 | . 2 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ∈ ℝ) |
9 | 4 | nnred 11674 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
10 | 2rp 12420 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ+) |
12 | 2z 12038 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
14 | 4 | nnzd 12110 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
15 | 13, 14 | zmulcld 12117 | . . . . 5 ⊢ (𝜑 → (2 · 𝑁) ∈ ℤ) |
16 | 11, 15 | rpexpcld 13643 | . . . 4 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ+) |
17 | 16 | rpred 12457 | . . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℝ) |
18 | 9, 17 | remulcld 10694 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ∈ ℝ) |
19 | fz1ssnn 12972 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ⊆ ℕ | |
20 | fzfi 13374 | . . . . 5 ⊢ (1...((2 · 𝑁) + 1)) ∈ Fin | |
21 | lcmfnncl 16010 | . . . . 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 11674 | . 2 ⊢ (𝜑 → (lcm‘(1...((2 · 𝑁) + 1))) ∈ ℝ) |
25 | lcmineqlem21.2 | . . . 4 ⊢ (𝜑 → 4 ≤ 𝑁) | |
26 | 4re 11743 | . . . . . 6 ⊢ 4 ∈ ℝ | |
27 | 26 | a1i 11 | . . . . 5 ⊢ (𝜑 → 4 ∈ ℝ) |
28 | 27, 9, 16 | lemul1d 12500 | . . . 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 11737 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
31 | 30, 2, 6 | expaddd 13547 | . . . . . 6 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = ((2↑(2 · 𝑁)) · (2↑2))) |
32 | sq2 13595 | . . . . . . 7 ⊢ (2↑2) = 4 | |
33 | 32 | oveq2i 7154 | . . . . . 6 ⊢ ((2↑(2 · 𝑁)) · (2↑2)) = ((2↑(2 · 𝑁)) · 4) |
34 | 31, 33 | eqtrdi 2810 | . . . . 5 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = ((2↑(2 · 𝑁)) · 4)) |
35 | 16 | rpcnd 12459 | . . . . . 6 ⊢ (𝜑 → (2↑(2 · 𝑁)) ∈ ℂ) |
36 | 27 | recnd 10692 | . . . . . 6 ⊢ (𝜑 → 4 ∈ ℂ) |
37 | 35, 36 | mulcomd 10685 | . . . . 5 ⊢ (𝜑 → ((2↑(2 · 𝑁)) · 4) = (4 · (2↑(2 · 𝑁)))) |
38 | 34, 37 | eqtrd 2794 | . . . 4 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) = (4 · (2↑(2 · 𝑁)))) |
39 | 38 | breq1d 5035 | . . 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 39600 | . 2 ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
42 | 8, 18, 24, 40, 41 | letrd 10820 | 1 ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ⊆ wss 3854 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 Fincfn 8520 ℝcr 10559 1c1 10561 + caddc 10563 · cmul 10565 ≤ cle 10699 ℕcn 11659 2c2 11714 4c4 11716 ℕ0cn0 11919 ℤcz 12005 ℝ+crp 12415 ...cfz 12924 ↑cexp 13464 lcmclcmf 15970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 ax-cc 9880 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 ax-addf 10639 ax-mulf 10640 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-symdif 4143 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-disj 4991 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-ofr 7399 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-fi 8893 df-sup 8924 df-inf 8925 df-oi 8992 df-dju 9348 df-card 9386 df-acn 9389 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-q 12374 df-rp 12416 df-xneg 12533 df-xadd 12534 df-xmul 12535 df-ioo 12768 df-ioc 12769 df-ico 12770 df-icc 12771 df-fz 12925 df-fzo 13068 df-fl 13196 df-mod 13272 df-seq 13404 df-exp 13465 df-fac 13669 df-bc 13698 df-hash 13726 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-limsup 14861 df-clim 14878 df-rlim 14879 df-sum 15076 df-prod 15293 df-dvds 15641 df-gcd 15879 df-lcm 15971 df-lcmf 15972 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-hom 16632 df-cco 16633 df-rest 16739 df-topn 16740 df-0g 16758 df-gsum 16759 df-topgen 16760 df-pt 16761 df-prds 16764 df-xrs 16818 df-qtop 16823 df-imas 16824 df-xps 16826 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-mulg 18277 df-cntz 18499 df-cmn 18960 df-psmet 20143 df-xmet 20144 df-met 20145 df-bl 20146 df-mopn 20147 df-fbas 20148 df-fg 20149 df-cnfld 20152 df-top 21579 df-topon 21596 df-topsp 21618 df-bases 21631 df-cld 21704 df-ntr 21705 df-cls 21706 df-nei 21783 df-lp 21821 df-perf 21822 df-cn 21912 df-cnp 21913 df-haus 22000 df-cmp 22072 df-tx 22247 df-hmeo 22440 df-fil 22531 df-fm 22623 df-flim 22624 df-flf 22625 df-xms 23007 df-ms 23008 df-tms 23009 df-cncf 23564 df-ovol 24149 df-vol 24150 df-mbf 24304 df-itg1 24305 df-itg2 24306 df-ibl 24307 df-itg 24308 df-0p 24355 df-limc 24550 df-dv 24551 |
This theorem is referenced by: lcmineqlem22 39602 |
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