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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 15142 | . . 3 ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ))) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))) |
3 | eleq2 2677 | . . . 4 ⊢ (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍)) | |
4 | raleq 3115 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛)) | |
5 | 4 | rabbidv 3164 | . . . . 5 ⊢ (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}) |
6 | 5 | infeq1d 8266 | . . . 4 ⊢ (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) |
7 | 3, 6 | ifbieq2d 4061 | . . 3 ⊢ (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, < ))) |
8 | 7 | adantl 481 | . 2 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 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, < ))) |
9 | zex 11263 | . . . . . 6 ⊢ ℤ ∈ V | |
10 | 9 | ssex 4730 | . . . . 5 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ V) |
11 | elpwg 4116 | . . . . 5 ⊢ (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ)) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ)) |
13 | 12 | ibir 256 | . . 3 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ) |
14 | 13 | adantr 480 | . 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑍 ∈ 𝒫 ℤ) |
15 | 0nn0 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
16 | 15 | a1i 11 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∈ 𝑍) → 0 ∈ ℕ0) |
17 | df-nel 2783 | . . . 4 ⊢ (0 ∉ 𝑍 ↔ ¬ 0 ∈ 𝑍) | |
18 | ssrab2 3650 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ ℕ | |
19 | nnssnn0 11172 | . . . . . 6 ⊢ ℕ ⊆ ℕ0 | |
20 | 18, 19 | sstri 3577 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ ℕ0 |
21 | nnuz 11599 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
22 | 18, 21 | sseqtri 3600 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ⊆ (ℤ≥‘1) |
23 | fissn0dvdsn0 15171 | . . . . . . 7 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅) | |
24 | 23 | 3expa 1257 | . . . . . 6 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅) |
25 | infssuzcl 11648 | . . . . . 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}) | |
26 | 22, 24, 25 | sylancr 694 | . . . . 5 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}) |
27 | 20, 26 | sseldi 3566 | . . . 4 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ 0 ∉ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ ℕ0) |
28 | 17, 27 | sylan2br 492 | . . 3 ⊢ (((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) ∧ ¬ 0 ∈ 𝑍) → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ) ∈ ℕ0) |
29 | 16, 28 | ifclda 4070 | . 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) ∈ ℕ0) |
30 | 2, 8, 14, 29 | fvmptd 6197 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 < clt 9953 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ∥ cdvds 14821 lcmclcmf 15140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 df-dvds 14822 df-lcmf 15142 |
This theorem is referenced by: lcmfn0val 15174 lcmfpr 15178 |
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