Proof of Theorem smndex1n0mnd
| Step | Hyp | Ref
| Expression |
| 1 | | smndex1ibas.n |
. . . . . . 7
⊢ 𝑁 ∈ ℕ |
| 2 | | nnnn0 12533 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
| 3 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑁 → (( I ↾
ℕ0)‘𝑥) = (( I ↾
ℕ0)‘𝑁)) |
| 4 | 1, 2 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ 𝑁 ∈
ℕ0 |
| 5 | | fvresi 7193 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (( I ↾ ℕ0)‘𝑁) = 𝑁) |
| 6 | 4, 5 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (( I
↾ ℕ0)‘𝑁) = 𝑁 |
| 7 | 3, 6 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → (( I ↾
ℕ0)‘𝑥) = 𝑁) |
| 8 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → (𝐼‘𝑥) = (𝐼‘𝑁)) |
| 9 | 7, 8 | eqeq12d 2753 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑁 → ((( I ↾
ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ 𝑁 = (𝐼‘𝑁))) |
| 10 | 9 | notbid 318 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (¬ (( I ↾
ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ 𝑁 = (𝐼‘𝑁))) |
| 11 | 10 | adantl 481 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑥 = 𝑁) → (¬ (( I ↾
ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ 𝑁 = (𝐼‘𝑁))) |
| 12 | | nnne0 12300 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) |
| 13 | 12 | neneqd 2945 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → ¬
𝑁 = 0) |
| 14 | | smndex1ibas.i |
. . . . . . . . . . 11
⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) |
| 15 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑁 → (𝑥 mod 𝑁) = (𝑁 mod 𝑁)) |
| 16 | | nnrp 13046 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ+) |
| 17 | | modid0 13937 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℝ+
→ (𝑁 mod 𝑁) = 0) |
| 18 | 16, 17 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → (𝑁 mod 𝑁) = 0) |
| 19 | 15, 18 | sylan9eqr 2799 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑥 = 𝑁) → (𝑥 mod 𝑁) = 0) |
| 20 | | c0ex 11255 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
| 21 | 20 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → 0 ∈
V) |
| 22 | 14, 19, 2, 21 | fvmptd2 7024 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (𝐼‘𝑁) = 0) |
| 23 | 22 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → (𝑁 = (𝐼‘𝑁) ↔ 𝑁 = 0)) |
| 24 | 13, 23 | mtbird 325 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → ¬
𝑁 = (𝐼‘𝑁)) |
| 25 | 2, 11, 24 | rspcedvd 3624 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ →
∃𝑥 ∈
ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥)) |
| 26 | 1, 25 | ax-mp 5 |
. . . . . 6
⊢
∃𝑥 ∈
ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) |
| 27 | | rexnal 3100 |
. . . . . 6
⊢
(∃𝑥 ∈
ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = (𝐼‘𝑥)) |
| 28 | 26, 27 | mpbi 230 |
. . . . 5
⊢ ¬
∀𝑥 ∈
ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) |
| 29 | | fnresi 6697 |
. . . . . 6
⊢ ( I
↾ ℕ0) Fn ℕ0 |
| 30 | | ovex 7464 |
. . . . . . 7
⊢ (𝑥 mod 𝑁) ∈ V |
| 31 | 30, 14 | fnmpti 6711 |
. . . . . 6
⊢ 𝐼 Fn
ℕ0 |
| 32 | | eqfnfv 7051 |
. . . . . 6
⊢ ((( I
↾ ℕ0) Fn ℕ0 ∧ 𝐼 Fn ℕ0) → (( I ↾
ℕ0) = 𝐼
↔ ∀𝑥 ∈
ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥))) |
| 33 | 29, 31, 32 | mp2an 692 |
. . . . 5
⊢ (( I
↾ ℕ0) = 𝐼 ↔ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = (𝐼‘𝑥)) |
| 34 | 28, 33 | mtbir 323 |
. . . 4
⊢ ¬ (
I ↾ ℕ0) = 𝐼 |
| 35 | 4 | a1i 11 |
. . . . . . . 8
⊢ (𝑛 ∈ (0..^𝑁) → 𝑁 ∈
ℕ0) |
| 36 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → ((𝐺‘𝑛)‘𝑥) = ((𝐺‘𝑛)‘𝑁)) |
| 37 | 7, 36 | eqeq12d 2753 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑁 → ((( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ 𝑁 = ((𝐺‘𝑛)‘𝑁))) |
| 38 | 37 | notbid 318 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (¬ (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁))) |
| 39 | 38 | adantl 481 |
. . . . . . . 8
⊢ ((𝑛 ∈ (0..^𝑁) ∧ 𝑥 = 𝑁) → (¬ (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁))) |
| 40 | | fzonel 13713 |
. . . . . . . . . . 11
⊢ ¬
𝑁 ∈ (0..^𝑁) |
| 41 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0..^𝑁) ↔ 𝑁 ∈ (0..^𝑁))) |
| 42 | 41 | eqcoms 2745 |
. . . . . . . . . . 11
⊢ (𝑁 = 𝑛 → (𝑛 ∈ (0..^𝑁) ↔ 𝑁 ∈ (0..^𝑁))) |
| 43 | 40, 42 | mtbiri 327 |
. . . . . . . . . 10
⊢ (𝑁 = 𝑛 → ¬ 𝑛 ∈ (0..^𝑁)) |
| 44 | 43 | con2i 139 |
. . . . . . . . 9
⊢ (𝑛 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑛) |
| 45 | | nn0ex 12532 |
. . . . . . . . . . . . 13
⊢
ℕ0 ∈ V |
| 46 | 45 | mptex 7243 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℕ0
↦ 𝑛) ∈
V |
| 47 | | smndex1ibas.g |
. . . . . . . . . . . . 13
⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) |
| 48 | 47 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ (0..^𝑁) ∧ (𝑥 ∈ ℕ0 ↦ 𝑛) ∈ V) → (𝐺‘𝑛) = (𝑥 ∈ ℕ0 ↦ 𝑛)) |
| 49 | 46, 48 | mpan2 691 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (0..^𝑁) → (𝐺‘𝑛) = (𝑥 ∈ ℕ0 ↦ 𝑛)) |
| 50 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ (0..^𝑁) ∧ 𝑥 = 𝑁) → 𝑛 = 𝑛) |
| 51 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ (0..^𝑁)) |
| 52 | 49, 50, 35, 51 | fvmptd 7023 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (0..^𝑁) → ((𝐺‘𝑛)‘𝑁) = 𝑛) |
| 53 | 52 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑛 ∈ (0..^𝑁) → (𝑁 = ((𝐺‘𝑛)‘𝑁) ↔ 𝑁 = 𝑛)) |
| 54 | 44, 53 | mtbird 325 |
. . . . . . . 8
⊢ (𝑛 ∈ (0..^𝑁) → ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁)) |
| 55 | 35, 39, 54 | rspcedvd 3624 |
. . . . . . 7
⊢ (𝑛 ∈ (0..^𝑁) → ∃𝑥 ∈ ℕ0 ¬ (( I
↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥)) |
| 56 | | rexnal 3100 |
. . . . . . 7
⊢
(∃𝑥 ∈
ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥)) |
| 57 | 55, 56 | sylib 218 |
. . . . . 6
⊢ (𝑛 ∈ (0..^𝑁) → ¬ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥)) |
| 58 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑛 ∈ V |
| 59 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℕ0
↦ 𝑛) = (𝑥 ∈ ℕ0
↦ 𝑛) |
| 60 | 58, 59 | fnmpti 6711 |
. . . . . . . 8
⊢ (𝑥 ∈ ℕ0
↦ 𝑛) Fn
ℕ0 |
| 61 | 49 | fneq1d 6661 |
. . . . . . . 8
⊢ (𝑛 ∈ (0..^𝑁) → ((𝐺‘𝑛) Fn ℕ0 ↔ (𝑥 ∈ ℕ0
↦ 𝑛) Fn
ℕ0)) |
| 62 | 60, 61 | mpbiri 258 |
. . . . . . 7
⊢ (𝑛 ∈ (0..^𝑁) → (𝐺‘𝑛) Fn ℕ0) |
| 63 | | eqfnfv 7051 |
. . . . . . 7
⊢ ((( I
↾ ℕ0) Fn ℕ0 ∧ (𝐺‘𝑛) Fn ℕ0) → (( I ↾
ℕ0) = (𝐺‘𝑛) ↔ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥))) |
| 64 | 29, 62, 63 | sylancr 587 |
. . . . . 6
⊢ (𝑛 ∈ (0..^𝑁) → (( I ↾ ℕ0) =
(𝐺‘𝑛) ↔ ∀𝑥 ∈ ℕ0 (( I ↾
ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥))) |
| 65 | 57, 64 | mtbird 325 |
. . . . 5
⊢ (𝑛 ∈ (0..^𝑁) → ¬ ( I ↾
ℕ0) = (𝐺‘𝑛)) |
| 66 | 65 | nrex 3074 |
. . . 4
⊢ ¬
∃𝑛 ∈ (0..^𝑁)( I ↾
ℕ0) = (𝐺‘𝑛) |
| 67 | 34, 66 | pm3.2ni 881 |
. . 3
⊢ ¬ ((
I ↾ ℕ0) = 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)) |
| 68 | | smndex1ibas.m |
. . . . . . . 8
⊢ 𝑀 =
(EndoFMnd‘ℕ0) |
| 69 | 68 | efmndid 18901 |
. . . . . . 7
⊢
(ℕ0 ∈ V → ( I ↾ ℕ0) =
(0g‘𝑀)) |
| 70 | 45, 69 | ax-mp 5 |
. . . . . 6
⊢ ( I
↾ ℕ0) = (0g‘𝑀) |
| 71 | 70 | eqcomi 2746 |
. . . . 5
⊢
(0g‘𝑀) = ( I ↾
ℕ0) |
| 72 | | smndex1mgm.b |
. . . . 5
⊢ 𝐵 = ({𝐼} ∪ ∪
𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) |
| 73 | 71, 72 | eleq12i 2834 |
. . . 4
⊢
((0g‘𝑀) ∈ 𝐵 ↔ ( I ↾ ℕ0)
∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) |
| 74 | | elun 4153 |
. . . . 5
⊢ (( I
↾ ℕ0) ∈ ({𝐼} ∪ ∪
𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0)
∈ {𝐼} ∨ ( I ↾
ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) |
| 75 | | resiexg 7934 |
. . . . . . . 8
⊢
(ℕ0 ∈ V → ( I ↾ ℕ0)
∈ V) |
| 76 | 45, 75 | ax-mp 5 |
. . . . . . 7
⊢ ( I
↾ ℕ0) ∈ V |
| 77 | 76 | elsn 4641 |
. . . . . 6
⊢ (( I
↾ ℕ0) ∈ {𝐼} ↔ ( I ↾ ℕ0) =
𝐼) |
| 78 | | eliun 4995 |
. . . . . . 7
⊢ (( I
↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) ∈
{(𝐺‘𝑛)}) |
| 79 | 76 | elsn 4641 |
. . . . . . . 8
⊢ (( I
↾ ℕ0) ∈ {(𝐺‘𝑛)} ↔ ( I ↾ ℕ0) =
(𝐺‘𝑛)) |
| 80 | 79 | rexbii 3094 |
. . . . . . 7
⊢
(∃𝑛 ∈
(0..^𝑁)( I ↾
ℕ0) ∈ {(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)) |
| 81 | 78, 80 | bitri 275 |
. . . . . 6
⊢ (( I
↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)) |
| 82 | 77, 81 | orbi12i 915 |
. . . . 5
⊢ ((( I
↾ ℕ0) ∈ {𝐼} ∨ ( I ↾ ℕ0)
∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0)
= 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))) |
| 83 | 74, 82 | bitri 275 |
. . . 4
⊢ (( I
↾ ℕ0) ∈ ({𝐼} ∪ ∪
𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0)
= 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))) |
| 84 | 73, 83 | bitri 275 |
. . 3
⊢
((0g‘𝑀) ∈ 𝐵 ↔ (( I ↾ ℕ0) =
𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))) |
| 85 | 67, 84 | mtbir 323 |
. 2
⊢ ¬
(0g‘𝑀)
∈ 𝐵 |
| 86 | 85 | nelir 3049 |
1
⊢
(0g‘𝑀) ∉ 𝐵 |