Step |
Hyp |
Ref |
Expression |
0 |
|
ce1 |
⊢ eval1 |
1 |
|
vr |
⊢ 𝑟 |
2 |
|
cvv |
⊢ V |
3 |
|
cbs |
⊢ Base |
4 |
1
|
cv |
⊢ 𝑟 |
5 |
4 3
|
cfv |
⊢ ( Base ‘ 𝑟 ) |
6 |
|
vb |
⊢ 𝑏 |
7 |
|
vx |
⊢ 𝑥 |
8 |
6
|
cv |
⊢ 𝑏 |
9 |
|
cmap |
⊢ ↑m |
10 |
|
c1o |
⊢ 1o |
11 |
8 10 9
|
co |
⊢ ( 𝑏 ↑m 1o ) |
12 |
8 11 9
|
co |
⊢ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) |
13 |
7
|
cv |
⊢ 𝑥 |
14 |
|
vy |
⊢ 𝑦 |
15 |
14
|
cv |
⊢ 𝑦 |
16 |
15
|
csn |
⊢ { 𝑦 } |
17 |
10 16
|
cxp |
⊢ ( 1o × { 𝑦 } ) |
18 |
14 8 17
|
cmpt |
⊢ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) |
19 |
13 18
|
ccom |
⊢ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) |
20 |
7 12 19
|
cmpt |
⊢ ( 𝑥 ∈ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) ↦ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) ) |
21 |
|
cevl |
⊢ eval |
22 |
10 4 21
|
co |
⊢ ( 1o eval 𝑟 ) |
23 |
20 22
|
ccom |
⊢ ( ( 𝑥 ∈ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) ↦ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) ) ∘ ( 1o eval 𝑟 ) ) |
24 |
6 5 23
|
csb |
⊢ ⦋ ( Base ‘ 𝑟 ) / 𝑏 ⦌ ( ( 𝑥 ∈ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) ↦ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) ) ∘ ( 1o eval 𝑟 ) ) |
25 |
1 2 24
|
cmpt |
⊢ ( 𝑟 ∈ V ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑏 ⦌ ( ( 𝑥 ∈ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) ↦ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) ) ∘ ( 1o eval 𝑟 ) ) ) |
26 |
0 25
|
wceq |
⊢ eval1 = ( 𝑟 ∈ V ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑏 ⦌ ( ( 𝑥 ∈ ( 𝑏 ↑m ( 𝑏 ↑m 1o ) ) ↦ ( 𝑥 ∘ ( 𝑦 ∈ 𝑏 ↦ ( 1o × { 𝑦 } ) ) ) ) ∘ ( 1o eval 𝑟 ) ) ) |