Step | Hyp | Ref
| Expression |
1 | | evl1fval.o |
. . 3
⊢ 𝑂 = (eval1‘𝑅) |
2 | | fvexd 6203 |
. . . . 5
⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V) |
3 | | id 22 |
. . . . . . . . 9
⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟)) |
4 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) |
5 | | evl1fval.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑅) |
6 | 4, 5 | syl6eqr 2674 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | 3, 6 | sylan9eqr 2678 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵) |
8 | 7 | oveq1d 6665 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑𝑚
1𝑜) = (𝐵
↑𝑚 1𝑜)) |
9 | 7, 8 | oveq12d 6668 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) = (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜))) |
10 | 7 | mpteq1d 4738 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})) = (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦}))) |
11 | 10 | coeq2d 5284 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) |
12 | 9, 11 | mpteq12dv 4733 |
. . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) = (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦}))))) |
13 | | simpl 473 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅) |
14 | 13 | oveq2d 6666 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = (1𝑜 eval
𝑅)) |
15 | | evl1fval.q |
. . . . . . 7
⊢ 𝑄 = (1𝑜 eval
𝑅) |
16 | 14, 15 | syl6eqr 2674 |
. . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = 𝑄) |
17 | 12, 16 | coeq12d 5286 |
. . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) ∘
(1𝑜 eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄)) |
18 | 2, 17 | csbied 3560 |
. . . 4
⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) ∘
(1𝑜 eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄)) |
19 | | df-evl1 19681 |
. . . 4
⊢
eval1 = (𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) ∘
(1𝑜 eval 𝑟))) |
20 | | ovex 6678 |
. . . . . 6
⊢ (𝐵 ↑𝑚
(𝐵
↑𝑚 1𝑜)) ∈ V |
21 | 20 | mptex 6486 |
. . . . 5
⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∈
V |
22 | | ovex 6678 |
. . . . . 6
⊢
(1𝑜 eval 𝑅) ∈ V |
23 | 15, 22 | eqeltri 2697 |
. . . . 5
⊢ 𝑄 ∈ V |
24 | 21, 23 | coex 7118 |
. . . 4
⊢ ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄) ∈ V |
25 | 18, 19, 24 | fvmpt 6282 |
. . 3
⊢ (𝑅 ∈ V →
(eval1‘𝑅)
= ((𝑥 ∈ (𝐵 ↑𝑚
(𝐵
↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄)) |
26 | 1, 25 | syl5eq 2668 |
. 2
⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄)) |
27 | | fvprc 6185 |
. . . . 5
⊢ (¬
𝑅 ∈ V →
(eval1‘𝑅)
= ∅) |
28 | 1, 27 | syl5eq 2668 |
. . . 4
⊢ (¬
𝑅 ∈ V → 𝑂 = ∅) |
29 | | co02 5649 |
. . . 4
⊢ ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ ∅) =
∅ |
30 | 28, 29 | syl6eqr 2674 |
. . 3
⊢ (¬
𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘
∅)) |
31 | | df-evl 19507 |
. . . . . . 7
⊢ eval =
(𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) |
32 | 31 | reldmmpt2 6771 |
. . . . . 6
⊢ Rel dom
eval |
33 | 32 | ovprc2 6685 |
. . . . 5
⊢ (¬
𝑅 ∈ V →
(1𝑜 eval 𝑅) = ∅) |
34 | 15, 33 | syl5eq 2668 |
. . . 4
⊢ (¬
𝑅 ∈ V → 𝑄 = ∅) |
35 | 34 | coeq2d 5284 |
. . 3
⊢ (¬
𝑅 ∈ V → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘
∅)) |
36 | 30, 35 | eqtr4d 2659 |
. 2
⊢ (¬
𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄)) |
37 | 26, 36 | pm2.61i 176 |
1
⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 ×
{𝑦})))) ∘ 𝑄) |