Step | Hyp | Ref
| Expression |
1 | | oveq2 6557 |
. . . . 5
⊢ (𝑥 = 𝑦 → (2𝑜
·𝑜 𝑥) = (2𝑜
·𝑜 𝑦)) |
2 | 1 | eqeq2d 2620 |
. . . 4
⊢ (𝑥 = 𝑦 → (𝐴 = (2𝑜
·𝑜 𝑥) ↔ 𝐴 = (2𝑜
·𝑜 𝑦))) |
3 | 2 | cbvrexv 3148 |
. . 3
⊢
(∃𝑥 ∈
ω 𝐴 =
(2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2𝑜
·𝑜 𝑦)) |
4 | | nnneo 7618 |
. . . . . . 7
⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2𝑜
·𝑜 𝑦)) → ¬ suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
5 | 4 | 3com23 1263 |
. . . . . 6
⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜
·𝑜 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
6 | 5 | 3expa 1257 |
. . . . 5
⊢ (((𝑦 ∈ ω ∧ 𝐴 = (2𝑜
·𝑜 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
7 | 6 | nrexdv 2984 |
. . . 4
⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜
·𝑜 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
8 | 7 | rexlimiva 3010 |
. . 3
⊢
(∃𝑦 ∈
ω 𝐴 =
(2𝑜 ·𝑜 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
9 | 3, 8 | sylbi 206 |
. 2
⊢
(∃𝑥 ∈
ω 𝐴 =
(2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥)) |
10 | | suceq 5707 |
. . . . . . 7
⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) |
11 | 10 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑦 = ∅ → (suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ suc ∅ = (2𝑜
·𝑜 𝑥))) |
12 | 11 | rexbidv 3034 |
. . . . 5
⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ =
(2𝑜 ·𝑜 𝑥))) |
13 | 12 | notbid 307 |
. . . 4
⊢ (𝑦 = ∅ → (¬
∃𝑥 ∈ ω suc
𝑦 = (2𝑜
·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ =
(2𝑜 ·𝑜 𝑥))) |
14 | | eqeq1 2614 |
. . . . 5
⊢ (𝑦 = ∅ → (𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∅ = (2𝑜
·𝑜 𝑥))) |
15 | 14 | rexbidv 3034 |
. . . 4
⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω ∅ =
(2𝑜 ·𝑜 𝑥))) |
16 | 13, 15 | imbi12d 333 |
. . 3
⊢ (𝑦 = ∅ → ((¬
∃𝑥 ∈ ω suc
𝑦 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc ∅ =
(2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ =
(2𝑜 ·𝑜 𝑥)))) |
17 | | suceq 5707 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧) |
18 | 17 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑦 = 𝑧 → (suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
19 | 18 | rexbidv 3034 |
. . . . 5
⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
20 | 19 | notbid 307 |
. . . 4
⊢ (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
21 | | eqeq1 2614 |
. . . . 5
⊢ (𝑦 = 𝑧 → (𝑦 = (2𝑜
·𝑜 𝑥) ↔ 𝑧 = (2𝑜
·𝑜 𝑥))) |
22 | 21 | rexbidv 3034 |
. . . 4
⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥))) |
23 | 20, 22 | imbi12d 333 |
. . 3
⊢ (𝑦 = 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥)))) |
24 | | suceq 5707 |
. . . . . . 7
⊢ (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧) |
25 | 24 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑦 = suc 𝑧 → (suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
26 | 25 | rexbidv 3034 |
. . . . 5
⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
27 | 26 | notbid 307 |
. . . 4
⊢ (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
28 | | eqeq1 2614 |
. . . . 5
⊢ (𝑦 = suc 𝑧 → (𝑦 = (2𝑜
·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
29 | 28 | rexbidv 3034 |
. . . 4
⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
30 | 27, 29 | imbi12d 333 |
. . 3
⊢ (𝑦 = suc 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥)))) |
31 | | suceq 5707 |
. . . . . . 7
⊢ (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴) |
32 | 31 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑦 = 𝐴 → (suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ suc 𝐴 = (2𝑜
·𝑜 𝑥))) |
33 | 32 | rexbidv 3034 |
. . . . 5
⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥))) |
34 | 33 | notbid 307 |
. . . 4
⊢ (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥))) |
35 | | eqeq1 2614 |
. . . . 5
⊢ (𝑦 = 𝐴 → (𝑦 = (2𝑜
·𝑜 𝑥) ↔ 𝐴 = (2𝑜
·𝑜 𝑥))) |
36 | 35 | rexbidv 3034 |
. . . 4
⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2𝑜
·𝑜 𝑥))) |
37 | 34, 36 | imbi12d 333 |
. . 3
⊢ (𝑦 = 𝐴 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜
·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜
·𝑜 𝑥)))) |
38 | | peano1 6977 |
. . . . 5
⊢ ∅
∈ ω |
39 | | eqid 2610 |
. . . . 5
⊢ ∅ =
∅ |
40 | | oveq2 6557 |
. . . . . . . 8
⊢ (𝑥 = ∅ →
(2𝑜 ·𝑜 𝑥) = (2𝑜
·𝑜 ∅)) |
41 | | om0x 7486 |
. . . . . . . 8
⊢
(2𝑜 ·𝑜 ∅) =
∅ |
42 | 40, 41 | syl6eq 2660 |
. . . . . . 7
⊢ (𝑥 = ∅ →
(2𝑜 ·𝑜 𝑥) = ∅) |
43 | 42 | eqeq2d 2620 |
. . . . . 6
⊢ (𝑥 = ∅ → (∅ =
(2𝑜 ·𝑜 𝑥) ↔ ∅ = ∅)) |
44 | 43 | rspcev 3282 |
. . . . 5
⊢ ((∅
∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ =
(2𝑜 ·𝑜 𝑥)) |
45 | 38, 39, 44 | mp2an 704 |
. . . 4
⊢
∃𝑥 ∈
ω ∅ = (2𝑜 ·𝑜 𝑥) |
46 | 45 | a1i 11 |
. . 3
⊢ (¬
∃𝑥 ∈ ω suc
∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ =
(2𝑜 ·𝑜 𝑥)) |
47 | 1 | eqeq2d 2620 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑧 = (2𝑜
·𝑜 𝑥) ↔ 𝑧 = (2𝑜
·𝑜 𝑦))) |
48 | 47 | cbvrexv 3148 |
. . . . . 6
⊢
(∃𝑥 ∈
ω 𝑧 =
(2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑦)) |
49 | | peano2 6978 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ω → suc 𝑦 ∈
ω) |
50 | | 2onn 7607 |
. . . . . . . . . . . 12
⊢
2𝑜 ∈ ω |
51 | | nnmsuc 7574 |
. . . . . . . . . . . 12
⊢
((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) →
(2𝑜 ·𝑜 suc 𝑦) = ((2𝑜
·𝑜 𝑦) +𝑜
2𝑜)) |
52 | 50, 51 | mpan 702 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω →
(2𝑜 ·𝑜 suc 𝑦) = ((2𝑜
·𝑜 𝑦) +𝑜
2𝑜)) |
53 | | df-2o 7448 |
. . . . . . . . . . . . 13
⊢
2𝑜 = suc 1𝑜 |
54 | 53 | oveq2i 6560 |
. . . . . . . . . . . 12
⊢
((2𝑜 ·𝑜 𝑦) +𝑜
2𝑜) = ((2𝑜 ·𝑜
𝑦) +𝑜
suc 1𝑜) |
55 | | nnmcl 7579 |
. . . . . . . . . . . . . 14
⊢
((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) →
(2𝑜 ·𝑜 𝑦) ∈ ω) |
56 | 50, 55 | mpan 702 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ω →
(2𝑜 ·𝑜 𝑦) ∈ ω) |
57 | | 1onn 7606 |
. . . . . . . . . . . . 13
⊢
1𝑜 ∈ ω |
58 | | nnasuc 7573 |
. . . . . . . . . . . . 13
⊢
(((2𝑜 ·𝑜 𝑦) ∈ ω ∧ 1𝑜
∈ ω) → ((2𝑜 ·𝑜
𝑦) +𝑜
suc 1𝑜) = suc ((2𝑜
·𝑜 𝑦) +𝑜
1𝑜)) |
59 | 56, 57, 58 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ω →
((2𝑜 ·𝑜 𝑦) +𝑜 suc
1𝑜) = suc ((2𝑜
·𝑜 𝑦) +𝑜
1𝑜)) |
60 | 54, 59 | syl5req 2657 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → suc
((2𝑜 ·𝑜 𝑦) +𝑜
1𝑜) = ((2𝑜 ·𝑜
𝑦) +𝑜
2𝑜)) |
61 | | nnon 6963 |
. . . . . . . . . . . 12
⊢
((2𝑜 ·𝑜 𝑦) ∈ ω →
(2𝑜 ·𝑜 𝑦) ∈ On) |
62 | | oa1suc 7498 |
. . . . . . . . . . . 12
⊢
((2𝑜 ·𝑜 𝑦) ∈ On → ((2𝑜
·𝑜 𝑦) +𝑜
1𝑜) = suc (2𝑜
·𝑜 𝑦)) |
63 | | suceq 5707 |
. . . . . . . . . . . 12
⊢
(((2𝑜 ·𝑜 𝑦) +𝑜
1𝑜) = suc (2𝑜
·𝑜 𝑦) → suc ((2𝑜
·𝑜 𝑦) +𝑜
1𝑜) = suc suc (2𝑜
·𝑜 𝑦)) |
64 | 56, 61, 62, 63 | 4syl 19 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → suc
((2𝑜 ·𝑜 𝑦) +𝑜
1𝑜) = suc suc (2𝑜
·𝑜 𝑦)) |
65 | 52, 60, 64 | 3eqtr2rd 2651 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ω → suc suc
(2𝑜 ·𝑜 𝑦) = (2𝑜
·𝑜 suc 𝑦)) |
66 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑥 = suc 𝑦 → (2𝑜
·𝑜 𝑥) = (2𝑜
·𝑜 suc 𝑦)) |
67 | 66 | eqeq2d 2620 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑦 → (suc suc (2𝑜
·𝑜 𝑦) = (2𝑜
·𝑜 𝑥) ↔ suc suc (2𝑜
·𝑜 𝑦) = (2𝑜
·𝑜 suc 𝑦))) |
68 | 67 | rspcev 3282 |
. . . . . . . . . 10
⊢ ((suc
𝑦 ∈ ω ∧ suc
suc (2𝑜 ·𝑜 𝑦) = (2𝑜
·𝑜 suc 𝑦)) → ∃𝑥 ∈ ω suc suc
(2𝑜 ·𝑜 𝑦) = (2𝑜
·𝑜 𝑥)) |
69 | 49, 65, 68 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω →
∃𝑥 ∈ ω suc
suc (2𝑜 ·𝑜 𝑦) = (2𝑜
·𝑜 𝑥)) |
70 | | suceq 5707 |
. . . . . . . . . . . 12
⊢ (𝑧 = (2𝑜
·𝑜 𝑦) → suc 𝑧 = suc (2𝑜
·𝑜 𝑦)) |
71 | | suceq 5707 |
. . . . . . . . . . . 12
⊢ (suc
𝑧 = suc
(2𝑜 ·𝑜 𝑦) → suc suc 𝑧 = suc suc (2𝑜
·𝑜 𝑦)) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑧 = (2𝑜
·𝑜 𝑦) → suc suc 𝑧 = suc suc (2𝑜
·𝑜 𝑦)) |
73 | 72 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ (𝑧 = (2𝑜
·𝑜 𝑦) → (suc suc 𝑧 = (2𝑜
·𝑜 𝑥) ↔ suc suc (2𝑜
·𝑜 𝑦) = (2𝑜
·𝑜 𝑥))) |
74 | 73 | rexbidv 3034 |
. . . . . . . . 9
⊢ (𝑧 = (2𝑜
·𝑜 𝑦) → (∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc
(2𝑜 ·𝑜 𝑦) = (2𝑜
·𝑜 𝑥))) |
75 | 69, 74 | syl5ibrcom 236 |
. . . . . . . 8
⊢ (𝑦 ∈ ω → (𝑧 = (2𝑜
·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
76 | 75 | rexlimiv 3009 |
. . . . . . 7
⊢
(∃𝑦 ∈
ω 𝑧 =
(2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥)) |
77 | 76 | a1i 11 |
. . . . . 6
⊢ (𝑧 ∈ ω →
(∃𝑦 ∈ ω
𝑧 = (2𝑜
·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
78 | 48, 77 | syl5bi 231 |
. . . . 5
⊢ (𝑧 ∈ ω →
(∃𝑥 ∈ ω
𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
79 | 78 | con3d 147 |
. . . 4
⊢ (𝑧 ∈ ω → (¬
∃𝑥 ∈ ω suc
suc 𝑧 =
(2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥))) |
80 | | con1 142 |
. . . 4
⊢ ((¬
∃𝑥 ∈ ω suc
𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥)) → (¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥))) |
81 | 79, 80 | syl9 75 |
. . 3
⊢ (𝑧 ∈ ω → ((¬
∃𝑥 ∈ ω suc
𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜
·𝑜 𝑥)) → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜
·𝑜 𝑥)))) |
82 | 16, 23, 30, 37, 46, 81 | finds 6984 |
. 2
⊢ (𝐴 ∈ ω → (¬
∃𝑥 ∈ ω suc
𝐴 = (2𝑜
·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜
·𝑜 𝑥))) |
83 | 9, 82 | impbid2 215 |
1
⊢ (𝐴 ∈ ω →
(∃𝑥 ∈ ω
𝐴 = (2𝑜
·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜
·𝑜 𝑥))) |