Step | Hyp | Ref
| Expression |
1 | | df-lcm 12015 |
. . 3
⊢ lcm =
(𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦
if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))) |
2 | 1 | a1i 9 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → lcm =
(𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦
if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))) |
3 | | eqeq1 2177 |
. . . . . 6
⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0)) |
4 | 3 | orbi1d 786 |
. . . . 5
⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0))) |
5 | | breq1 3992 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑀 ∥ 𝑛)) |
6 | 5 | anbi1d 462 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → ((𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛))) |
7 | 6 | rabbidv 2719 |
. . . . . 6
⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}) |
8 | 7 | infeq1d 6989 |
. . . . 5
⊢ (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) |
9 | 4, 8 | ifbieq2d 3550 |
. . . 4
⊢ (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, < ))) |
10 | | eqeq1 2177 |
. . . . . 6
⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0)) |
11 | 10 | orbi2d 785 |
. . . . 5
⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0))) |
12 | | breq1 3992 |
. . . . . . . 8
⊢ (𝑦 = 𝑁 → (𝑦 ∥ 𝑛 ↔ 𝑁 ∥ 𝑛)) |
13 | 12 | anbi2d 461 |
. . . . . . 7
⊢ (𝑦 = 𝑁 → ((𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
14 | 13 | rabbidv 2719 |
. . . . . 6
⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
15 | 14 | infeq1d 6989 |
. . . . 5
⊢ (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) |
16 | 11, 15 | ifbieq2d 3550 |
. . . 4
⊢ (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, < ))) |
17 | 9, 16 | sylan9eq 2223 |
. . 3
⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
18 | 17 | adantl 275 |
. 2
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 = 𝑀 ∧ 𝑦 = 𝑁)) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |
19 | | simpl 108 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈
ℤ) |
20 | | simpr 109 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈
ℤ) |
21 | | c0ex 7914 |
. . . 4
⊢ 0 ∈
V |
22 | 21 | a1i 9 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 0 ∈ V) |
23 | | 1zzd 9239 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 1 ∈
ℤ) |
24 | | nnuz 9522 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
25 | | rabeq 2722 |
. . . . . 6
⊢ (ℕ
= (ℤ≥‘1) → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} = {𝑛 ∈ (ℤ≥‘1)
∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
26 | 24, 25 | ax-mp 5 |
. . . . 5
⊢ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} = {𝑛 ∈ (ℤ≥‘1)
∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} |
27 | | dvdsmul1 11775 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁)) |
28 | 27 | adantr 274 |
. . . . . . 7
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (𝑀 · 𝑁)) |
29 | | simpll 524 |
. . . . . . . 8
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℤ) |
30 | | simplr 525 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ) |
31 | 29, 30 | zmulcld 9340 |
. . . . . . . 8
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 · 𝑁) ∈ ℤ) |
32 | | dvdsabsb 11772 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁)))) |
33 | 29, 31, 32 | syl2anc 409 |
. . . . . . 7
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁)))) |
34 | 28, 33 | mpbid 146 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (abs‘(𝑀 · 𝑁))) |
35 | | dvdsmul2 11776 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
36 | 35 | adantr 274 |
. . . . . . 7
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (𝑀 · 𝑁)) |
37 | | dvdsabsb 11772 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))) |
38 | 30, 31, 37 | syl2anc 409 |
. . . . . . 7
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))) |
39 | 36, 38 | mpbid 146 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (abs‘(𝑀 · 𝑁))) |
40 | 29 | zcnd 9335 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℂ) |
41 | 30 | zcnd 9335 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℂ) |
42 | 40, 41 | absmuld 11158 |
. . . . . . . 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 747 |
. . . . . . . . . . . . 13
⊢ (¬
(𝑀 = 0 ∨ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0)) |
45 | 43, 44 | sylib 121 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0)) |
46 | 45 | simpld 111 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑀 = 0) |
47 | 46 | neqned 2347 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ≠ 0) |
48 | | nnabscl 11064 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈
ℕ) |
49 | 29, 47, 48 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑀) ∈
ℕ) |
50 | 45 | simprd 113 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑁 = 0) |
51 | 50 | neqned 2347 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ≠ 0) |
52 | | nnabscl 11064 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈
ℕ) |
53 | 30, 51, 52 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑁) ∈
ℕ) |
54 | 49, 53 | nnmulcld 8927 |
. . . . . . . 8
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) · (abs‘𝑁)) ∈
ℕ) |
55 | 42, 54 | eqeltrd 2247 |
. . . . . . 7
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ ℕ) |
56 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁)))) |
57 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑁 ∥ 𝑛 ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))) |
58 | 56, 57 | anbi12d 470 |
. . . . . . . 8
⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))) |
59 | 58 | elrab3 2887 |
. . . . . . 7
⊢
((abs‘(𝑀
· 𝑁)) ∈ ℕ
→ ((abs‘(𝑀
· 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))) |
60 | 55, 59 | syl 14 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))) |
61 | 34, 39, 60 | mpbir2and 939 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
62 | | elfzelz 9981 |
. . . . . . 7
⊢ (𝑛 ∈ (1...(abs‘(𝑀 · 𝑁))) → 𝑛 ∈ ℤ) |
63 | | zdvdsdc 11774 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) →
DECID 𝑀
∥ 𝑛) |
64 | 29, 62, 63 | syl2an 287 |
. . . . . 6
⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑀 ∥ 𝑛) |
65 | | zdvdsdc 11774 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) →
DECID 𝑁
∥ 𝑛) |
66 | 30, 62, 65 | syl2an 287 |
. . . . . 6
⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑁 ∥ 𝑛) |
67 | | dcan2 929 |
. . . . . 6
⊢
(DECID 𝑀 ∥ 𝑛 → (DECID 𝑁 ∥ 𝑛 → DECID (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
68 | 64, 66, 67 | sylc 62 |
. . . . 5
⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)) |
69 | 23, 26, 61, 68 | infssuzcldc 11906 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) |
70 | 69 | elexd 2743 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈
V) |
71 | | lcmmndc 12016 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
DECID (𝑀 = 0
∨ 𝑁 =
0)) |
72 | 22, 70, 71 | ifcldadc 3555 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) ∈
V) |
73 | 2, 18, 19, 20, 72 | ovmpod 5980 |
1
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))) |