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| Description: The least common multiple of the empty set is 1. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) | 
| Ref | Expression | 
|---|---|
| lcmf0 | ⊢ (lcm‘∅) = 1 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ ℤ | |
| 2 | 0fi 9082 | . . 3 ⊢ ∅ ∈ Fin | |
| 3 | noel 4338 | . . . 4 ⊢ ¬ 0 ∈ ∅ | |
| 4 | 3 | nelir 3049 | . . 3 ⊢ 0 ∉ ∅ | 
| 5 | lcmfn0val 16660 | . . 3 ⊢ ((∅ ⊆ ℤ ∧ ∅ ∈ Fin ∧ 0 ∉ ∅) → (lcm‘∅) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛}, ℝ, < )) | |
| 6 | 1, 2, 4, 5 | mp3an 1463 | . 2 ⊢ (lcm‘∅) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛}, ℝ, < ) | 
| 7 | ral0 4513 | . . . . . 6 ⊢ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛 | |
| 8 | 7 | rgenw 3065 | . . . . 5 ⊢ ∀𝑛 ∈ ℕ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛 | 
| 9 | rabid2 3470 | . . . . 5 ⊢ (ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛} ↔ ∀𝑛 ∈ ℕ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛) | |
| 10 | 8, 9 | mpbir 231 | . . . 4 ⊢ ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛} | 
| 11 | 10 | eqcomi 2746 | . . 3 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛} = ℕ | 
| 12 | 11 | infeq1i 9518 | . 2 ⊢ inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑛}, ℝ, < ) = inf(ℕ, ℝ, < ) | 
| 13 | nninf 12971 | . 2 ⊢ inf(ℕ, ℝ, < ) = 1 | |
| 14 | 6, 12, 13 | 3eqtri 2769 | 1 ⊢ (lcm‘∅) = 1 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 {crab 3436 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 Fincfn 8985 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 < clt 11295 ℕcn 12266 ℤcz 12613 ∥ 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: lcmfunsnlem 16678 lcmfun 16682 lcm1un 42014 | 
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