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Mirrors > Home > MPE Home > Th. List > lcmval | Structured version Visualization version GIF version |
Description: Value of the lcm operator. (𝑀 lcm 𝑁) is the least common multiple of 𝑀 and 𝑁. If either 𝑀 or 𝑁 is 0, the result is defined conventionally as 0. Contrast with df-gcd 16529 and gcdval 16530. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
Ref | Expression |
---|---|
lcmval | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2739 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0)) | |
2 | 1 | orbi1d 916 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0))) |
3 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑀 ∥ 𝑛)) | |
4 | 3 | anbi1d 631 | . . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛))) |
5 | 4 | rabbidv 3441 | . . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}) |
6 | 5 | infeq1d 9515 | . . 3 ⊢ (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) |
7 | 2, 6 | ifbieq2d 4557 | . 2 ⊢ (𝑥 = 𝑀 → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))) |
8 | eqeq1 2739 | . . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0)) | |
9 | 8 | orbi2d 915 | . . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0))) |
10 | breq1 5151 | . . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑦 ∥ 𝑛 ↔ 𝑁 ∥ 𝑛)) | |
11 | 10 | anbi2d 630 | . . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
12 | 11 | rabbidv 3441 | . . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
13 | 12 | infeq1d 9515 | . . 3 ⊢ (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) |
14 | 9, 13 | ifbieq2d 4557 | . 2 ⊢ (𝑦 = 𝑁 → if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
15 | df-lcm 16624 | . 2 ⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))) | |
16 | c0ex 11253 | . . 3 ⊢ 0 ∈ V | |
17 | ltso 11339 | . . . 4 ⊢ < Or ℝ | |
18 | 17 | infex 9531 | . . 3 ⊢ inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ V |
19 | 16, 18 | ifex 4581 | . 2 ⊢ if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) ∈ V |
20 | 7, 14, 15, 19 | ovmpo 7593 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 class class class wbr 5148 (class class class)co 7431 infcinf 9479 ℝcr 11152 0cc0 11153 < clt 11293 ℕcn 12264 ℤcz 12611 ∥ cdvds 16287 lcm clcm 16622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-lcm 16624 |
This theorem is referenced by: lcmcom 16627 lcm0val 16628 lcmn0val 16629 lcmass 16648 lcmfpr 16661 |
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