Proof of Theorem lcmfunsnlem2
| Step | Hyp | Ref
| Expression |
| 1 | | nfv 1914 |
. . 3
⊢
Ⅎ𝑛(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) |
| 2 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑛∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) |
| 3 | | nfra1 3284 |
. . . 4
⊢
Ⅎ𝑛∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) |
| 4 | 2, 3 | nfan 1899 |
. . 3
⊢
Ⅎ𝑛(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) |
| 5 | 1, 4 | nfan 1899 |
. 2
⊢
Ⅎ𝑛((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧
(∀𝑘 ∈ ℤ
(∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) |
| 6 | | 0z 12624 |
. . . . 5
⊢ 0 ∈
ℤ |
| 7 | | eqoreldif 4685 |
. . . . 5
⊢ (0 ∈
ℤ → (𝑛 ∈
ℤ ↔ (𝑛 = 0 ∨
𝑛 ∈ (ℤ ∖
{0})))) |
| 8 | 6, 7 | ax-mp 5 |
. . . 4
⊢ (𝑛 ∈ ℤ ↔ (𝑛 = 0 ∨ 𝑛 ∈ (ℤ ∖
{0}))) |
| 9 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆
ℤ) |
| 10 | | snssi 4808 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ℤ → {𝑧} ⊆
ℤ) |
| 11 | 10 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆
ℤ) |
| 12 | 9, 11 | unssd 4192 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ) |
| 13 | | snssi 4808 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℤ → {0} ⊆ ℤ) |
| 14 | 6, 13 | mp1i 13 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {0} ⊆
ℤ) |
| 15 | 12, 14 | unssd 4192 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {0}) ⊆
ℤ) |
| 16 | | c0ex 11255 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
| 17 | 16 | snid 4662 |
. . . . . . . . . . . . 13
⊢ 0 ∈
{0} |
| 18 | 17 | olci 867 |
. . . . . . . . . . . 12
⊢ (0 ∈
(𝑦 ∪ {𝑧}) ∨ 0 ∈
{0}) |
| 19 | | elun 4153 |
. . . . . . . . . . . 12
⊢ (0 ∈
((𝑦 ∪ {𝑧}) ∪ {0}) ↔ (0 ∈
(𝑦 ∪ {𝑧}) ∨ 0 ∈
{0})) |
| 20 | 18, 19 | mpbir 231 |
. . . . . . . . . . 11
⊢ 0 ∈
((𝑦 ∪ {𝑧}) ∪ {0}) |
| 21 | | lcmf0val 16659 |
. . . . . . . . . . 11
⊢ ((((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ ∧ 0 ∈
((𝑦 ∪ {𝑧}) ∪ {0})) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {0})) =
0) |
| 22 | 15, 20, 21 | sylancl 586 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {0})) =
0) |
| 23 | 22 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {0})) =
0) |
| 24 | | sneq 4636 |
. . . . . . . . . . . 12
⊢ (𝑛 = 0 → {𝑛} = {0}) |
| 25 | 24 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → {𝑛} = {0}) |
| 26 | 25 | uneq2d 4168 |
. . . . . . . . . 10
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {0})) |
| 27 | 26 | fveq2d 6910 |
. . . . . . . . 9
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {0}))) |
| 28 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑛 = 0 →
((lcm‘(𝑦 ∪
{𝑧})) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 0)) |
| 29 | | snfi 9083 |
. . . . . . . . . . . . . . 15
⊢ {𝑧} ∈ Fin |
| 30 | | unfi 9211 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin) |
| 31 | 29, 30 | mpan2 691 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin) |
| 32 | 31 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin) |
| 33 | | lcmfcl 16665 |
. . . . . . . . . . . . 13
⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) →
(lcm‘(𝑦 ∪
{𝑧})) ∈
ℕ0) |
| 34 | 12, 32, 33 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
(lcm‘(𝑦 ∪
{𝑧})) ∈
ℕ0) |
| 35 | 34 | nn0zd 12639 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
(lcm‘(𝑦 ∪
{𝑧})) ∈
ℤ) |
| 36 | | lcm0val 16631 |
. . . . . . . . . . 11
⊢
((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ →
((lcm‘(𝑦 ∪
{𝑧})) lcm 0) =
0) |
| 37 | 35, 36 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
((lcm‘(𝑦 ∪
{𝑧})) lcm 0) =
0) |
| 38 | 28, 37 | sylan9eqr 2799 |
. . . . . . . . 9
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) →
((lcm‘(𝑦 ∪
{𝑧})) lcm 𝑛) = 0) |
| 39 | 23, 27, 38 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)) |
| 40 | 39 | ex 412 |
. . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 = 0 →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 41 | 40 | adantr 480 |
. . . . . 6
⊢ (((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 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 42 | 41 | com12 32 |
. . . . 5
⊢ (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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 43 | 9 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
𝑦 ⊆
ℤ) |
| 44 | 11 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
{𝑧} ⊆
ℤ) |
| 45 | 43, 44 | unssd 4192 |
. . . . . . . . . . . . 13
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(𝑦 ∪ {𝑧}) ⊆
ℤ) |
| 46 | | elun1 4182 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
𝑦 → 0 ∈ (𝑦 ∪ {𝑧})) |
| 47 | 46 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
0 ∈ (𝑦 ∪ {𝑧})) |
| 48 | | lcmf0val 16659 |
. . . . . . . . . . . . 13
⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ 0 ∈ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) = 0) |
| 49 | 45, 47, 48 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(lcm‘(𝑦 ∪
{𝑧})) = 0) |
| 50 | 49 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(𝑛 lcm
(lcm‘(𝑦 ∪
{𝑧}))) = (𝑛 lcm 0)) |
| 51 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℤ ∖ {0})
→ 𝑛 ∈
ℤ) |
| 52 | | lcm0val 16631 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℤ → (𝑛 lcm 0) = 0) |
| 53 | 51, 52 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (ℤ ∖ {0})
→ (𝑛 lcm 0) =
0) |
| 54 | 53 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(𝑛 lcm 0) =
0) |
| 55 | 50, 54 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(𝑛 lcm
(lcm‘(𝑦 ∪
{𝑧}))) =
0) |
| 56 | | simp3 1139 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin) |
| 57 | 56, 29, 30 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin) |
| 58 | 12, 57, 33 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
(lcm‘(𝑦 ∪
{𝑧})) ∈
ℕ0) |
| 59 | 58 | nn0zd 12639 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) →
(lcm‘(𝑦 ∪
{𝑧})) ∈
ℤ) |
| 60 | 51 | adantl 481 |
. . . . . . . . . . 11
⊢ ((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
→ 𝑛 ∈
ℤ) |
| 61 | | lcmcom 16630 |
. . . . . . . . . . 11
⊢
(((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) →
((lcm‘(𝑦 ∪
{𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧})))) |
| 62 | 59, 60, 61 | syl2anr 597 |
. . . . . . . . . 10
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
((lcm‘(𝑦 ∪
{𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧})))) |
| 63 | 12 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(𝑦 ∪ {𝑧}) ⊆
ℤ) |
| 64 | 51 | snssd 4809 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℤ ∖ {0})
→ {𝑛} ⊆
ℤ) |
| 65 | 64 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
{𝑛} ⊆
ℤ) |
| 66 | 63, 65 | unssd 4192 |
. . . . . . . . . . 11
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ) |
| 67 | 46 | orcd 874 |
. . . . . . . . . . . . 13
⊢ (0 ∈
𝑦 → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛})) |
| 68 | | elun 4153 |
. . . . . . . . . . . . 13
⊢ (0 ∈
((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛})) |
| 69 | 67, 68 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (0 ∈
𝑦 → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) |
| 70 | 69 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) |
| 71 | | lcmf0val 16659 |
. . . . . . . . . . 11
⊢ ((((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0) |
| 72 | 66, 70, 71 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = 0) |
| 73 | 55, 62, 72 | 3eqtr4rd 2788 |
. . . . . . . . 9
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0}))
∧ (𝑧 ∈ ℤ
∧ 𝑦 ⊆ ℤ
∧ 𝑦 ∈ Fin)) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)) |
| 74 | 73 | a1d 25 |
. . . . . . . 8
⊢ (((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {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)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 75 | 74 | expimpd 453 |
. . . . . . 7
⊢ ((0
∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 76 | 75 | ex 412 |
. . . . . 6
⊢ (0 ∈
𝑦 → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))) |
| 77 | | elsng 4640 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 ∈
ℤ → (0 ∈ {𝑧} ↔ 0 = 𝑧)) |
| 78 | | eqcom 2744 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 =
𝑧 ↔ 𝑧 = 0) |
| 79 | 77, 78 | bitrdi 287 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
ℤ → (0 ∈ {𝑧} ↔ 𝑧 = 0)) |
| 80 | 6, 79 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
{𝑧} ↔ 𝑧 = 0) |
| 81 | 80 | biimpri 228 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 0 → 0 ∈ {𝑧}) |
| 82 | 81 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈
{𝑧}) |
| 83 | 82 | olcd 875 |
. . . . . . . . . . . . . 14
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈
𝑦 ∨ 0 ∈ {𝑧})) |
| 84 | | elun 4153 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
(𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧})) |
| 85 | 83, 84 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈
(𝑦 ∪ {𝑧})) |
| 86 | 12, 85, 48 | syl2an2 686 |
. . . . . . . . . . . 12
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) →
(lcm‘(𝑦 ∪
{𝑧})) = 0) |
| 87 | 86 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0)) |
| 88 | 51 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑛 ∈
ℤ) |
| 89 | 88, 52 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0) |
| 90 | 87, 89 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0) |
| 91 | 59, 88, 61 | syl2an2 686 |
. . . . . . . . . 10
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) →
((lcm‘(𝑦 ∪
{𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧})))) |
| 92 | 12 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ) |
| 93 | 64 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆
ℤ) |
| 94 | 92, 93 | unssd 4192 |
. . . . . . . . . . 11
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ) |
| 95 | | sneq 4636 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 0 → {𝑧} = {0}) |
| 96 | 17, 95 | eleqtrrid 2848 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 0 → 0 ∈ {𝑧}) |
| 97 | 96 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈
{𝑧}) |
| 98 | 97 | olcd 875 |
. . . . . . . . . . . . . 14
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈
𝑦 ∨ 0 ∈ {𝑧})) |
| 99 | 98, 84 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈
(𝑦 ∪ {𝑧})) |
| 100 | 99 | orcd 874 |
. . . . . . . . . . . 12
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈
(𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛})) |
| 101 | 100, 68 | sylibr 234 |
. . . . . . . . . . 11
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈
((𝑦 ∪ {𝑧}) ∪ {𝑛})) |
| 102 | 94, 101, 71 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = 0) |
| 103 | 90, 91, 102 | 3eqtr4rd 2788 |
. . . . . . . . 9
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)) |
| 104 | 103 | a1d 25 |
. . . . . . . 8
⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {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)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 105 | 104 | expimpd 453 |
. . . . . . 7
⊢ ((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 106 | 105 | ex 412 |
. . . . . 6
⊢ (𝑧 = 0 → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))) |
| 107 | | ioran 986 |
. . . . . . . 8
⊢ (¬ (0
∈ 𝑦 ∨ 𝑧 = 0) ↔ (¬ 0 ∈
𝑦 ∧ ¬ 𝑧 = 0)) |
| 108 | | df-nel 3047 |
. . . . . . . . 9
⊢ (0
∉ 𝑦 ↔ ¬ 0
∈ 𝑦) |
| 109 | | df-ne 2941 |
. . . . . . . . 9
⊢ (𝑧 ≠ 0 ↔ ¬ 𝑧 = 0) |
| 110 | 108, 109 | anbi12i 628 |
. . . . . . . 8
⊢ ((0
∉ 𝑦 ∧ 𝑧 ≠ 0) ↔ (¬ 0 ∈
𝑦 ∧ ¬ 𝑧 = 0)) |
| 111 | 107, 110 | bitr4i 278 |
. . . . . . 7
⊢ (¬ (0
∈ 𝑦 ∨ 𝑧 = 0) ↔ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0)) |
| 112 | | eldif 3961 |
. . . . . . . 8
⊢ (𝑛 ∈ (ℤ ∖ {0})
↔ (𝑛 ∈ ℤ
∧ ¬ 𝑛 ∈
{0})) |
| 113 | | velsn 4642 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ {0} ↔ 𝑛 = 0) |
| 114 | 113 | bicomi 224 |
. . . . . . . . . . 11
⊢ (𝑛 = 0 ↔ 𝑛 ∈ {0}) |
| 115 | 114 | necon3abii 2987 |
. . . . . . . . . 10
⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 ∈ {0}) |
| 116 | | lcmfunsnlem2lem2 16676 |
. . . . . . . . . . . 12
⊢ (((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))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)) |
| 117 | 116 | exp520 1358 |
. . . . . . . . . . 11
⊢ (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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))) |
| 118 | 117 | imp 406 |
. . . . . . . . . 10
⊢ ((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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))) |
| 119 | 115, 118 | biimtrrid 243 |
. . . . . . . . 9
⊢ ((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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))) |
| 120 | 119 | impcomd 411 |
. . . . . . . 8
⊢ ((0
∉ 𝑦 ∧ 𝑧 ≠ 0) → ((𝑛 ∈ ℤ ∧ ¬
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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))) |
| 121 | 112, 120 | biimtrid 242 |
. . . . . . 7
⊢ ((0
∉ 𝑦 ∧ 𝑧 ≠ 0) → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))) |
| 122 | 111, 121 | sylbi 217 |
. . . . . 6
⊢ (¬ (0
∈ 𝑦 ∨ 𝑧 = 0) → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))) |
| 123 | 76, 106, 122 | ecase3 1033 |
. . . . 5
⊢ (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 124 | 42, 123 | jaoi 858 |
. . . 4
⊢ ((𝑛 = 0 ∨ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 125 | 8, 124 | sylbi 217 |
. . 3
⊢ (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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 126 | 125 | com12 32 |
. 2
⊢ (((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 →
(lcm‘((𝑦 ∪
{𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))) |
| 127 | 5, 126 | ralrimi 3257 |
1
⊢ (((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 (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)) |