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| Mirrors > Home > MPE Home > Th. List > lcmfval | Structured version Visualization version GIF version | ||
| Description: Value of the lcm function. (lcm‘𝑍) is the least common multiple of the integers contained in the finite subset of integers 𝑍. If at least one of the elements of 𝑍 is 0, the result is defined conventionally as 0. (Contributed by AV, 21-Apr-2020.) (Revised by AV, 16-Sep-2020.) |
| Ref | Expression |
|---|---|
| lcmfval | ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-lcmf 16628 | . 2 ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ))) | |
| 2 | eleq2 2830 | . . 3 ⊢ (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍)) | |
| 3 | raleq 3323 | . . . . 5 ⊢ (𝑧 = 𝑍 → (∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛)) | |
| 4 | 3 | rabbidv 3444 | . . . 4 ⊢ (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}) |
| 5 | 4 | infeq1d 9517 | . . 3 ⊢ (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) |
| 6 | 2, 5 | ifbieq2d 4552 | . 2 ⊢ (𝑧 = 𝑍 → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))) |
| 7 | zex 12622 | . . . . . 6 ⊢ ℤ ∈ V | |
| 8 | 7 | ssex 5321 | . . . . 5 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ V) |
| 9 | elpwg 4603 | . . . . 5 ⊢ (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ)) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ)) |
| 11 | 10 | ibir 268 | . . 3 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ) |
| 12 | 11 | adantr 480 | . 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑍 ∈ 𝒫 ℤ) |
| 13 | 0nn0 12541 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 14 | 13 | a1i 11 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∈ 𝑍) → 0 ∈ ℕ0) |
| 15 | df-nel 3047 | . . . 4 ⊢ (0 ∉ 𝑍 ↔ ¬ 0 ∈ 𝑍) | |
| 16 | ssrab2 4080 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ ℕ | |
| 17 | nnssnn0 12529 | . . . . . 6 ⊢ ℕ ⊆ ℕ0 | |
| 18 | 16, 17 | sstri 3993 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ ℕ0 |
| 19 | nnuz 12921 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
| 20 | 16, 19 | sseqtri 4032 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ (ℤ≥‘1) |
| 21 | fissn0dvdsn0 16657 | . . . . . . 7 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅) | |
| 22 | 21 | 3expa 1119 | . . . . . 6 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅) |
| 23 | infssuzcl 12974 | . . . . . 6 ⊢ (({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ (ℤ≥‘1) ∧ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}) | |
| 24 | 20, 22, 23 | sylancr 587 | . . . . 5 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}) |
| 25 | 18, 24 | sselid 3981 | . . . 4 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ ℕ0) |
| 26 | 15, 25 | sylan2br 595 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ ¬ 0 ∈ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ ℕ0) |
| 27 | 14, 26 | ifclda 4561 | . 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) ∈ ℕ0) |
| 28 | 1, 6, 12, 27 | fvmptd3 7039 | 1 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ifcif 4525 𝒫 cpw 4600 class class class wbr 5143 ‘cfv 6561 Fincfn 8985 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 < clt 11295 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ∥ cdvds 16290 lcmclcmf 16626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-lcmf 16628 |
| This theorem is referenced by: lcmfn0val 16660 lcmfpr 16664 |
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