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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 16309 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → (lcm‘𝑍) = inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < )) | |
2 | 1 | adantr 480 | . . 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 4017 | . . . . . 6 ⊢ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ⊆ ℕ | |
4 | nnuz 12603 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
5 | 3, 4 | sseqtri 3961 | . . . . 5 ⊢ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ⊆ (ℤ≥‘1) |
6 | simpr 484 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) | |
7 | breq2 5082 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝐾)) | |
8 | 7 | ralbidv 3122 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) |
9 | 8 | elrab 3625 | . . . . . 6 ⊢ (𝐾 ∈ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘} ↔ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) |
10 | 6, 9 | sylibr 233 | . . . . 5 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → 𝐾 ∈ {𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}) |
11 | infssuzle 12653 | . . . . 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 586 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → inf({𝑘 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘}, ℝ, < ) ≤ 𝐾) |
13 | 12 | 3ad2antl1 1183 | . . 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 5100 | . 2 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) ∧ (𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) → (lcm‘𝑍) ≤ 𝐾) |
15 | 14 | ex 412 | 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 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∉ wnel 3050 ∀wral 3065 {crab 3069 ⊆ wss 3891 class class class wbr 5078 ‘cfv 6430 Fincfn 8707 infcinf 9161 ℝcr 10854 0cc0 10855 1c1 10856 < clt 10993 ≤ cle 10994 ℕcn 11956 ℤcz 12302 ℤ≥cuz 12564 ∥ cdvds 15944 lcmclcmf 16275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-prod 15597 df-dvds 15945 df-lcmf 16277 |
This theorem is referenced by: lcmf 16319 lcmflefac 16334 |
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