ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lcmval GIF version

Theorem lcmval 11955
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 11842 and gcdval 11858. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmval ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
Distinct variable groups:   𝑛,𝑀   𝑛,𝑁

Proof of Theorem lcmval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lcm 11953 . . 3 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
21a1i 9 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ))))
3 eqeq1 2164 . . . . . 6 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
43orbi1d 781 . . . . 5 (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
5 breq1 3970 . . . . . . . 8 (𝑥 = 𝑀 → (𝑥𝑛𝑀𝑛))
65anbi1d 461 . . . . . . 7 (𝑥 = 𝑀 → ((𝑥𝑛𝑦𝑛) ↔ (𝑀𝑛𝑦𝑛)))
76rabbidv 2701 . . . . . 6 (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)})
87infeq1d 6958 . . . . 5 (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ))
94, 8ifbieq2d 3530 . . . 4 (𝑥 = 𝑀 → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < )))
10 eqeq1 2164 . . . . . 6 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
1110orbi2d 780 . . . . 5 (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
12 breq1 3970 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦𝑛𝑁𝑛))
1312anbi2d 460 . . . . . . 7 (𝑦 = 𝑁 → ((𝑀𝑛𝑦𝑛) ↔ (𝑀𝑛𝑁𝑛)))
1413rabbidv 2701 . . . . . 6 (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
1514infeq1d 6958 . . . . 5 (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ))
1611, 15ifbieq2d 3530 . . . 4 (𝑦 = 𝑁 → if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
179, 16sylan9eq 2210 . . 3 ((𝑥 = 𝑀𝑦 = 𝑁) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
1817adantl 275 . 2 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 = 𝑀𝑦 = 𝑁)) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
19 simpl 108 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
20 simpr 109 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
21 c0ex 7874 . . . 4 0 ∈ V
2221a1i 9 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 0 ∈ V)
23 1zzd 9199 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 1 ∈ ℤ)
24 nnuz 9479 . . . . . 6 ℕ = (ℤ‘1)
25 rabeq 2704 . . . . . 6 (ℕ = (ℤ‘1) → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)} = {𝑛 ∈ (ℤ‘1) ∣ (𝑀𝑛𝑁𝑛)})
2624, 25ax-mp 5 . . . . 5 {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)} = {𝑛 ∈ (ℤ‘1) ∣ (𝑀𝑛𝑁𝑛)}
27 dvdsmul1 11720 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁))
2827adantr 274 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (𝑀 · 𝑁))
29 simpll 519 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℤ)
30 simplr 520 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
3129, 30zmulcld 9297 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 · 𝑁) ∈ ℤ)
32 dvdsabsb 11717 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
3329, 31, 32syl2anc 409 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
3428, 33mpbid 146 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (abs‘(𝑀 · 𝑁)))
35 dvdsmul2 11721 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))
3635adantr 274 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (𝑀 · 𝑁))
37 dvdsabsb 11717 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))
3830, 31, 37syl2anc 409 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))
3936, 38mpbid 146 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (abs‘(𝑀 · 𝑁)))
4029zcnd 9292 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℂ)
4130zcnd 9292 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℂ)
4240, 41absmuld 11105 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
43 simpr 109 . . . . . . . . . . . . 13 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
44 ioran 742 . . . . . . . . . . . . 13 (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0))
4543, 44sylib 121 . . . . . . . . . . . 12 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0))
4645simpld 111 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑀 = 0)
4746neqned 2334 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ≠ 0)
48 nnabscl 11011 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
4929, 47, 48syl2anc 409 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑀) ∈ ℕ)
5045simprd 113 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑁 = 0)
5150neqned 2334 . . . . . . . . . 10 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ≠ 0)
52 nnabscl 11011 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
5330, 51, 52syl2anc 409 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑁) ∈ ℕ)
5449, 53nnmulcld 8887 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) · (abs‘𝑁)) ∈ ℕ)
5542, 54eqeltrd 2234 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
56 breq2 3971 . . . . . . . . 9 (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑀𝑛𝑀 ∥ (abs‘(𝑀 · 𝑁))))
57 breq2 3971 . . . . . . . . 9 (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑁𝑛𝑁 ∥ (abs‘(𝑀 · 𝑁))))
5856, 57anbi12d 465 . . . . . . . 8 (𝑛 = (abs‘(𝑀 · 𝑁)) → ((𝑀𝑛𝑁𝑛) ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
5958elrab3 2869 . . . . . . 7 ((abs‘(𝑀 · 𝑁)) ∈ ℕ → ((abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
6055, 59syl 14 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
6134, 39, 60mpbir2and 929 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
62 elfzelz 9934 . . . . . . 7 (𝑛 ∈ (1...(abs‘(𝑀 · 𝑁))) → 𝑛 ∈ ℤ)
63 zdvdsdc 11719 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 𝑀𝑛)
6429, 62, 63syl2an 287 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑀𝑛)
65 zdvdsdc 11719 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 𝑁𝑛)
6630, 62, 65syl2an 287 . . . . . 6 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑁𝑛)
67 dcan 919 . . . . . 6 (DECID 𝑀𝑛 → (DECID 𝑁𝑛DECID (𝑀𝑛𝑁𝑛)))
6864, 66, 67sylc 62 . . . . 5 ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID (𝑀𝑛𝑁𝑛))
6923, 26, 61, 68infssuzcldc 11850 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
7069elexd 2725 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ V)
71 lcmmndc 11954 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
7222, 70, 71ifcldadc 3535 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) ∈ V)
732, 18, 19, 20, 72ovmpod 5950 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820   = wceq 1335  wcel 2128  wne 2327  {crab 2439  Vcvv 2712  ifcif 3506   class class class wbr 3967  cfv 5172  (class class class)co 5826  cmpo 5828  infcinf 6929  cr 7733  0cc0 7734  1c1 7735   · cmul 7739   < clt 7914  cn 8838  cz 9172  cuz 9444  ...cfz 9918  abscabs 10908  cdvds 11694   lcm clcm 11952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-isom 5181  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-sup 6930  df-inf 6931  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-fl 10178  df-mod 10231  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-dvds 11695  df-lcm 11953
This theorem is referenced by:  lcmcom  11956  lcm0val  11957  lcmn0val  11958  lcmass  11977
  Copyright terms: Public domain W3C validator