| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nfv |
⊢ Ⅎ 𝑘 ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) |
| 2 |
|
nfv |
⊢ Ⅎ 𝑘 𝑛 ∈ ℤ |
| 3 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) |
| 4 |
|
nfra1 |
⊢ Ⅎ 𝑘 ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) |
| 5 |
|
nfv |
⊢ Ⅎ 𝑘 ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) |
| 6 |
4 5
|
nfan |
⊢ Ⅎ 𝑘 ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) |
| 7 |
3 6
|
nfan |
⊢ Ⅎ 𝑘 ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) |
| 8 |
2 7
|
nfan |
⊢ Ⅎ 𝑘 ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) |
| 9 |
1 8
|
nfan |
⊢ Ⅎ 𝑘 ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) |
| 10 |
|
simprr |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → 𝑘 ∈ ℕ ) |
| 11 |
|
simp2 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → 𝑦 ⊆ ℤ ) |
| 12 |
|
snssi |
⊢ ( 𝑧 ∈ ℤ → { 𝑧 } ⊆ ℤ ) |
| 13 |
12
|
3ad2ant1 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → { 𝑧 } ⊆ ℤ ) |
| 14 |
11 13
|
unssd |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ) |
| 15 |
|
simp3 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → 𝑦 ∈ Fin ) |
| 16 |
|
snfi |
⊢ { 𝑧 } ∈ Fin |
| 17 |
|
unfi |
⊢ ( ( 𝑦 ∈ Fin ∧ { 𝑧 } ∈ Fin ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 18 |
15 16 17
|
sylancl |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 19 |
14 18
|
jca |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ∧ ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) ) |
| 20 |
|
lcmfcl |
⊢ ( ( ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ∧ ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℕ0 ) |
| 21 |
19 20
|
syl |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℕ0 ) |
| 22 |
21
|
nn0zd |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ) |
| 23 |
22
|
adantl |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ) |
| 24 |
23
|
adantr |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ) |
| 25 |
|
simprl |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → 𝑛 ∈ ℤ ) |
| 26 |
10 24 25
|
3jca |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ) |
| 27 |
14
|
adantl |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) → ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ) |
| 28 |
18
|
adantl |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 29 |
|
df-nel |
⊢ ( 0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦 ) |
| 30 |
29
|
biimpi |
⊢ ( 0 ∉ 𝑦 → ¬ 0 ∈ 𝑦 ) |
| 31 |
|
elsni |
⊢ ( 0 ∈ { 𝑧 } → 0 = 𝑧 ) |
| 32 |
31
|
eqcomd |
⊢ ( 0 ∈ { 𝑧 } → 𝑧 = 0 ) |
| 33 |
32
|
necon3ai |
⊢ ( 𝑧 ≠ 0 → ¬ 0 ∈ { 𝑧 } ) |
| 34 |
30 33
|
anim12i |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ) → ( ¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ { 𝑧 } ) ) |
| 35 |
34
|
3adant3 |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ { 𝑧 } ) ) |
| 36 |
|
df-nel |
⊢ ( 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ↔ ¬ 0 ∈ ( 𝑦 ∪ { 𝑧 } ) ) |
| 37 |
|
ioran |
⊢ ( ¬ ( 0 ∈ 𝑦 ∨ 0 ∈ { 𝑧 } ) ↔ ( ¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ { 𝑧 } ) ) |
| 38 |
|
elun |
⊢ ( 0 ∈ ( 𝑦 ∪ { 𝑧 } ) ↔ ( 0 ∈ 𝑦 ∨ 0 ∈ { 𝑧 } ) ) |
| 39 |
37 38
|
xchnxbir |
⊢ ( ¬ 0 ∈ ( 𝑦 ∪ { 𝑧 } ) ↔ ( ¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ { 𝑧 } ) ) |
| 40 |
36 39
|
bitri |
⊢ ( 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ↔ ( ¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ { 𝑧 } ) ) |
| 41 |
35 40
|
sylibr |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ) |
| 42 |
41
|
adantr |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) → 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ) |
| 43 |
27 28 42
|
3jca |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ∧ ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ∧ 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ) ) |
| 44 |
43
|
adantr |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ∧ ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ∧ 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ) ) |
| 45 |
|
lcmfn0cl |
⊢ ( ( ( 𝑦 ∪ { 𝑧 } ) ⊆ ℤ ∧ ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ∧ 0 ∉ ( 𝑦 ∪ { 𝑧 } ) ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℕ ) |
| 46 |
44 45
|
syl |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℕ ) |
| 47 |
46
|
nnne0d |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ≠ 0 ) |
| 48 |
47
|
neneqd |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ¬ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ) |
| 49 |
|
neneq |
⊢ ( 𝑛 ≠ 0 → ¬ 𝑛 = 0 ) |
| 50 |
49
|
3ad2ant3 |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ¬ 𝑛 = 0 ) |
| 51 |
50
|
ad2antrr |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ¬ 𝑛 = 0 ) |
| 52 |
48 51
|
jca |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( ¬ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∧ ¬ 𝑛 = 0 ) ) |
| 53 |
|
ioran |
⊢ ( ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ↔ ( ¬ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∧ ¬ 𝑛 = 0 ) ) |
| 54 |
52 53
|
sylibr |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) |
| 55 |
26 54
|
jca |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) |
| 56 |
55
|
exp43 |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( 𝑛 ∈ ℤ → ( 𝑘 ∈ ℕ → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) ) ) ) |
| 57 |
56
|
adantrd |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( 𝑛 ∈ ℤ → ( 𝑘 ∈ ℕ → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) ) ) ) |
| 58 |
57
|
com23 |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( 𝑛 ∈ ℤ → ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( 𝑘 ∈ ℕ → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) ) ) ) |
| 59 |
58
|
imp32 |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) → ( 𝑘 ∈ ℕ → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) ) |
| 60 |
59
|
imp |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) |
| 61 |
60
|
adantr |
⊢ ( ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) ) |
| 62 |
|
sneq |
⊢ ( 𝑛 = 𝑧 → { 𝑛 } = { 𝑧 } ) |
| 63 |
62
|
uneq2d |
⊢ ( 𝑛 = 𝑧 → ( 𝑦 ∪ { 𝑛 } ) = ( 𝑦 ∪ { 𝑧 } ) ) |
| 64 |
63
|
fveq2d |
⊢ ( 𝑛 = 𝑧 → ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ) |
| 65 |
|
oveq2 |
⊢ ( 𝑛 = 𝑧 → ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ) |
| 66 |
64 65
|
eqeq12d |
⊢ ( 𝑛 = 𝑧 → ( ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ↔ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ) ) |
| 67 |
66
|
rspcv |
⊢ ( 𝑧 ∈ ℤ → ( ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ) ) |
| 68 |
67
|
3ad2ant1 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ) ) |
| 69 |
|
nnz |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℤ ) |
| 70 |
69
|
adantl |
⊢ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℤ ) |
| 71 |
70
|
adantl |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → 𝑘 ∈ ℤ ) |
| 72 |
|
lcmfcl |
⊢ ( ( 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( lcm ‘ 𝑦 ) ∈ ℕ0 ) |
| 73 |
72
|
nn0zd |
⊢ ( ( 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( lcm ‘ 𝑦 ) ∈ ℤ ) |
| 74 |
73
|
3adant1 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( lcm ‘ 𝑦 ) ∈ ℤ ) |
| 75 |
74
|
ad2antrr |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( lcm ‘ 𝑦 ) ∈ ℤ ) |
| 76 |
|
simpll1 |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → 𝑧 ∈ ℤ ) |
| 77 |
71 75 76
|
3jca |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) → ( 𝑘 ∈ ℤ ∧ ( lcm ‘ 𝑦 ) ∈ ℤ ∧ 𝑧 ∈ ℤ ) ) |
| 78 |
77
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( 𝑘 ∈ ℤ ∧ ( lcm ‘ 𝑦 ) ∈ ℤ ∧ 𝑧 ∈ ℤ ) ) |
| 79 |
|
elun1 |
⊢ ( 𝑚 ∈ 𝑦 → 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) |
| 80 |
79
|
orcd |
⊢ ( 𝑚 ∈ 𝑦 → ( 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑚 ∈ { 𝑛 } ) ) |
| 81 |
|
elun |
⊢ ( 𝑚 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) ↔ ( 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑚 ∈ { 𝑛 } ) ) |
| 82 |
80 81
|
sylibr |
⊢ ( 𝑚 ∈ 𝑦 → 𝑚 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) ) |
| 83 |
|
breq1 |
⊢ ( 𝑖 = 𝑚 → ( 𝑖 ∥ 𝑘 ↔ 𝑚 ∥ 𝑘 ) ) |
| 84 |
83
|
rspcv |
⊢ ( 𝑚 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑚 ∥ 𝑘 ) ) |
| 85 |
82 84
|
syl |
⊢ ( 𝑚 ∈ 𝑦 → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑚 ∥ 𝑘 ) ) |
| 86 |
85
|
com12 |
⊢ ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( 𝑚 ∈ 𝑦 → 𝑚 ∥ 𝑘 ) ) |
| 87 |
86
|
adantl |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( 𝑚 ∈ 𝑦 → 𝑚 ∥ 𝑘 ) ) |
| 88 |
87
|
ralrimiv |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ) |
| 89 |
88
|
adantr |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) ∧ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ) → ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ) |
| 90 |
|
breq2 |
⊢ ( 𝑘 = 𝑙 → ( 𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙 ) ) |
| 91 |
90
|
ralbidv |
⊢ ( 𝑘 = 𝑙 → ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ) ) |
| 92 |
|
breq2 |
⊢ ( 𝑘 = 𝑙 → ( ( lcm ‘ 𝑦 ) ∥ 𝑘 ↔ ( lcm ‘ 𝑦 ) ∥ 𝑙 ) ) |
| 93 |
91 92
|
imbi12d |
⊢ ( 𝑘 = 𝑙 → ( ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ↔ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → ( lcm ‘ 𝑦 ) ∥ 𝑙 ) ) ) |
| 94 |
93
|
cbvralvw |
⊢ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ↔ ∀ 𝑙 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → ( lcm ‘ 𝑦 ) ∥ 𝑙 ) ) |
| 95 |
70
|
adantr |
⊢ ( ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → 𝑘 ∈ ℤ ) |
| 96 |
95
|
adantl |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) ∧ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ) → 𝑘 ∈ ℤ ) |
| 97 |
|
breq2 |
⊢ ( 𝑙 = 𝑘 → ( 𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘 ) ) |
| 98 |
97
|
ralbidv |
⊢ ( 𝑙 = 𝑘 → ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ) ) |
| 99 |
|
breq2 |
⊢ ( 𝑙 = 𝑘 → ( ( lcm ‘ 𝑦 ) ∥ 𝑙 ↔ ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) |
| 100 |
98 99
|
imbi12d |
⊢ ( 𝑙 = 𝑘 → ( ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → ( lcm ‘ 𝑦 ) ∥ 𝑙 ) ↔ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) |
| 101 |
100
|
rspcv |
⊢ ( 𝑘 ∈ ℤ → ( ∀ 𝑙 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → ( lcm ‘ 𝑦 ) ∥ 𝑙 ) → ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) |
| 102 |
96 101
|
syl |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) ∧ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ) → ( ∀ 𝑙 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → ( lcm ‘ 𝑦 ) ∥ 𝑙 ) → ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) |
| 103 |
94 102
|
biimtrid |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) ∧ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) |
| 104 |
89 103
|
mpid |
⊢ ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) ∧ ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) |
| 105 |
104
|
exp31 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) ) |
| 106 |
105
|
com24 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) ) |
| 107 |
106
|
imp |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) → ( ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ) |
| 108 |
107
|
impl |
⊢ ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) |
| 109 |
108
|
imp |
⊢ ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) |
| 110 |
|
vsnid |
⊢ 𝑧 ∈ { 𝑧 } |
| 111 |
110
|
olci |
⊢ ( 𝑧 ∈ 𝑦 ∨ 𝑧 ∈ { 𝑧 } ) |
| 112 |
|
elun |
⊢ ( 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) ↔ ( 𝑧 ∈ 𝑦 ∨ 𝑧 ∈ { 𝑧 } ) ) |
| 113 |
111 112
|
mpbir |
⊢ 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) |
| 114 |
113
|
orci |
⊢ ( 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑧 ∈ { 𝑛 } ) |
| 115 |
|
elun |
⊢ ( 𝑧 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) ↔ ( 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑧 ∈ { 𝑛 } ) ) |
| 116 |
114 115
|
mpbir |
⊢ 𝑧 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) |
| 117 |
|
breq1 |
⊢ ( 𝑖 = 𝑧 → ( 𝑖 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘 ) ) |
| 118 |
117
|
rspcv |
⊢ ( 𝑧 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑧 ∥ 𝑘 ) ) |
| 119 |
116 118
|
mp1i |
⊢ ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑧 ∥ 𝑘 ) ) |
| 120 |
119
|
imp |
⊢ ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → 𝑧 ∥ 𝑘 ) |
| 121 |
109 120
|
jca |
⊢ ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( ( lcm ‘ 𝑦 ) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘 ) ) |
| 122 |
|
lcmdvds |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( lcm ‘ 𝑦 ) ∈ ℤ ∧ 𝑧 ∈ ℤ ) → ( ( ( lcm ‘ 𝑦 ) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘 ) → ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ∥ 𝑘 ) ) |
| 123 |
78 121 122
|
sylc |
⊢ ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ∥ 𝑘 ) |
| 124 |
|
breq1 |
⊢ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ↔ ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) ∥ 𝑘 ) ) |
| 125 |
123 124
|
imbitrrid |
⊢ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) → ( ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) |
| 126 |
125
|
expd |
⊢ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) → ( ( ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) ) ∧ ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) |
| 127 |
126
|
exp5j |
⊢ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) → ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) ) |
| 128 |
127
|
com12 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑧 ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) ) |
| 129 |
68 128
|
syld |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) ) |
| 130 |
129
|
com23 |
⊢ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) → ( ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) → ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) ) |
| 131 |
130
|
imp32 |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( ( 𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) |
| 132 |
131
|
expd |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( 𝑛 ∈ ℤ → ( 𝑘 ∈ ℕ → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) |
| 133 |
132
|
com34 |
⊢ ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( 𝑛 ∈ ℤ → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) |
| 134 |
133
|
com12 |
⊢ ( 𝑛 ∈ ℤ → ( ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) ) |
| 135 |
134
|
imp |
⊢ ( ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) → ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) |
| 136 |
135
|
com12 |
⊢ ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) → ( ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) ) |
| 137 |
136
|
imp |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) ) |
| 138 |
137
|
imp |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) ) |
| 139 |
138
|
imp |
⊢ ( ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ) |
| 140 |
|
vsnid |
⊢ 𝑛 ∈ { 𝑛 } |
| 141 |
140
|
olci |
⊢ ( 𝑛 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑛 ∈ { 𝑛 } ) |
| 142 |
|
elun |
⊢ ( 𝑛 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) ↔ ( 𝑛 ∈ ( 𝑦 ∪ { 𝑧 } ) ∨ 𝑛 ∈ { 𝑛 } ) ) |
| 143 |
141 142
|
mpbir |
⊢ 𝑛 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) |
| 144 |
|
breq1 |
⊢ ( 𝑖 = 𝑛 → ( 𝑖 ∥ 𝑘 ↔ 𝑛 ∥ 𝑘 ) ) |
| 145 |
144
|
rspcv |
⊢ ( 𝑛 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑛 ∥ 𝑘 ) ) |
| 146 |
143 145
|
mp1i |
⊢ ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → 𝑛 ∥ 𝑘 ) ) |
| 147 |
146
|
imp |
⊢ ( ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → 𝑛 ∥ 𝑘 ) |
| 148 |
139 147
|
jca |
⊢ ( ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ∧ 𝑛 ∥ 𝑘 ) ) |
| 149 |
|
lcmledvds |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) ∧ ¬ ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) = 0 ∨ 𝑛 = 0 ) ) → ( ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) ∥ 𝑘 ∧ 𝑛 ∥ 𝑘 ) → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) lcm 𝑛 ) ≤ 𝑘 ) ) |
| 150 |
61 148 149
|
sylc |
⊢ ( ( ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) ∧ 𝑘 ∈ ℕ ) ∧ ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 ) → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) lcm 𝑛 ) ≤ 𝑘 ) |
| 151 |
150
|
exp31 |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) → ( 𝑘 ∈ ℕ → ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) lcm 𝑛 ) ≤ 𝑘 ) ) ) |
| 152 |
9 151
|
ralrimi |
⊢ ( ( ( 0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0 ) ∧ ( 𝑛 ∈ ℤ ∧ ( ( 𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ) ∧ ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑦 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑦 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑦 ) lcm 𝑛 ) ) ) ) ) → ∀ 𝑘 ∈ ℕ ( ∀ 𝑖 ∈ ( ( 𝑦 ∪ { 𝑧 } ) ∪ { 𝑛 } ) 𝑖 ∥ 𝑘 → ( ( lcm ‘ ( 𝑦 ∪ { 𝑧 } ) ) lcm 𝑛 ) ≤ 𝑘 ) ) |