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Mirrors > Home > MPE Home > Th. List > lcmflefac | Structured version Visualization version GIF version |
Description: The least common multiple of all positive integers less than or equal to an integer is less than or equal to the factorial of the integer. (Contributed by AV, 16-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
lcmflefac | ⊢ (𝑁 ∈ ℕ → (lcm‘(1...𝑁)) ≤ (!‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzssz 12912 | . . . 4 ⊢ (1...𝑁) ⊆ ℤ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ⊆ ℤ) |
3 | fzfid 13344 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin) | |
4 | 0nelfz1 12929 | . . . 4 ⊢ 0 ∉ (1...𝑁) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 ∉ (1...𝑁)) |
6 | 2, 3, 5 | 3jca 1124 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁))) |
7 | nnnn0 11907 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
8 | 7 | faccld 13647 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ) |
9 | elfznn 12939 | . . . . 5 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ) | |
10 | elfzuz3 12908 | . . . . . 6 ⊢ (𝑚 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘𝑚)) | |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑚)) |
12 | dvdsfac 15678 | . . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝑚)) → 𝑚 ∥ (!‘𝑁)) | |
13 | 9, 11, 12 | syl2an2 684 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∥ (!‘𝑁)) |
14 | 13 | ralrimiva 3184 | . . 3 ⊢ (𝑁 ∈ ℕ → ∀𝑚 ∈ (1...𝑁)𝑚 ∥ (!‘𝑁)) |
15 | 8, 14 | jca 514 | . 2 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) ∈ ℕ ∧ ∀𝑚 ∈ (1...𝑁)𝑚 ∥ (!‘𝑁))) |
16 | lcmfledvds 15978 | . 2 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (((!‘𝑁) ∈ ℕ ∧ ∀𝑚 ∈ (1...𝑁)𝑚 ∥ (!‘𝑁)) → (lcm‘(1...𝑁)) ≤ (!‘𝑁))) | |
17 | 6, 15, 16 | sylc 65 | 1 ⊢ (𝑁 ∈ ℕ → (lcm‘(1...𝑁)) ≤ (!‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∉ wnel 3125 ∀wral 3140 ⊆ wss 3938 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 0cc0 10539 1c1 10540 ≤ cle 10678 ℕcn 11640 ℤcz 11984 ℤ≥cuz 12246 ...cfz 12895 !cfa 13636 ∥ cdvds 15609 lcmclcmf 15935 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 df-dvds 15610 df-lcmf 15937 |
This theorem is referenced by: (None) |
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