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Mirrors > Home > MPE Home > Th. List > lcmfledvds | Structured version Visualization version GIF version |
Description: A positive integer which is divisible by all elements of a set of integers bounds the least common multiple of the set. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
Ref | Expression |
---|---|
lcmfledvds | ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → ((𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾) → (lcm‘𝑍) ≤ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmfn0val 15967 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → (lcm‘𝑍) = inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < )) | |
2 | 1 | adantr 483 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → (lcm‘𝑍) = inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < )) |
3 | ssrab2 4056 | . . . . . 6 ⊢ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ⊆ ℕ | |
4 | nnuz 12282 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
5 | 3, 4 | sseqtri 4003 | . . . . 5 ⊢ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ⊆ (ℤ≥‘1) |
6 | simpr 487 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) | |
7 | breq2 5070 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝐾)) | |
8 | 7 | ralbidv 3197 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) |
9 | 8 | elrab 3680 | . . . . . 6 ⊢ (𝐾 ∈ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ↔ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) |
10 | 6, 9 | sylibr 236 | . . . . 5 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → 𝐾 ∈ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}) |
11 | infssuzle 12332 | . . . . 5 ⊢ (({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ⊆ (ℤ≥‘1) ∧ 𝐾 ∈ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}) → inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < ) ≤ 𝐾) | |
12 | 5, 10, 11 | sylancr 589 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < ) ≤ 𝐾) |
13 | 12 | 3ad2antl1 1181 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < ) ≤ 𝐾) |
14 | 2, 13 | eqbrtrd 5088 | . 2 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → (lcm‘𝑍) ≤ 𝐾) |
15 | 14 | ex 415 | 1 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → ((𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾) → (lcm‘𝑍) ≤ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 ∀wral 3138 {crab 3142 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 Fincfn 8509 infcinf 8905 ℝcr 10536 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 ∥ cdvds 15607 lcmclcmf 15933 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-lcmf 15935 |
This theorem is referenced by: lcmf 15977 lcmflefac 15992 |
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