Step |
Hyp |
Ref |
Expression |
1 |
|
efgval.w |
⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) ) |
2 |
|
efgval.r |
⊢ ∼ = ( ~FG ‘ 𝐼 ) |
3 |
|
efgval2.m |
⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ 〈 𝑦 , ( 1o ∖ 𝑧 ) 〉 ) |
4 |
|
efgval2.t |
⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) ) |
5 |
|
fviss |
⊢ ( I ‘ Word ( 𝐼 × 2o ) ) ⊆ Word ( 𝐼 × 2o ) |
6 |
1 5
|
eqsstri |
⊢ 𝑊 ⊆ Word ( 𝐼 × 2o ) |
7 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → 𝑋 ∈ 𝑊 ) |
8 |
6 7
|
sselid |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → 𝑋 ∈ Word ( 𝐼 × 2o ) ) |
9 |
|
simprr |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → 𝑏 ∈ ( 𝐼 × 2o ) ) |
10 |
3
|
efgmf |
⊢ 𝑀 : ( 𝐼 × 2o ) ⟶ ( 𝐼 × 2o ) |
11 |
10
|
ffvelrni |
⊢ ( 𝑏 ∈ ( 𝐼 × 2o ) → ( 𝑀 ‘ 𝑏 ) ∈ ( 𝐼 × 2o ) ) |
12 |
11
|
ad2antll |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → ( 𝑀 ‘ 𝑏 ) ∈ ( 𝐼 × 2o ) ) |
13 |
9 12
|
s2cld |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 ∈ Word ( 𝐼 × 2o ) ) |
14 |
|
splcl |
⊢ ( ( 𝑋 ∈ Word ( 𝐼 × 2o ) ∧ 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 ∈ Word ( 𝐼 × 2o ) ) → ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ∈ Word ( 𝐼 × 2o ) ) |
15 |
8 13 14
|
syl2anc |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ∈ Word ( 𝐼 × 2o ) ) |
16 |
1
|
efgrcl |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝐼 ∈ V ∧ 𝑊 = Word ( 𝐼 × 2o ) ) ) |
17 |
16
|
simprd |
⊢ ( 𝑋 ∈ 𝑊 → 𝑊 = Word ( 𝐼 × 2o ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → 𝑊 = Word ( 𝐼 × 2o ) ) |
19 |
15 18
|
eleqtrrd |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∧ 𝑏 ∈ ( 𝐼 × 2o ) ) ) → ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ∈ 𝑊 ) |
20 |
19
|
ralrimivva |
⊢ ( 𝑋 ∈ 𝑊 → ∀ 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∀ 𝑏 ∈ ( 𝐼 × 2o ) ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ∈ 𝑊 ) |
21 |
|
eqid |
⊢ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) |
22 |
21
|
fmpo |
⊢ ( ∀ 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∀ 𝑏 ∈ ( 𝐼 × 2o ) ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ∈ 𝑊 ↔ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ) |
23 |
20 22
|
sylib |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ) |
24 |
|
ovex |
⊢ ( 0 ... ( ♯ ‘ 𝑋 ) ) ∈ V |
25 |
16
|
simpld |
⊢ ( 𝑋 ∈ 𝑊 → 𝐼 ∈ V ) |
26 |
|
2on |
⊢ 2o ∈ On |
27 |
|
xpexg |
⊢ ( ( 𝐼 ∈ V ∧ 2o ∈ On ) → ( 𝐼 × 2o ) ∈ V ) |
28 |
25 26 27
|
sylancl |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝐼 × 2o ) ∈ V ) |
29 |
|
xpexg |
⊢ ( ( ( 0 ... ( ♯ ‘ 𝑋 ) ) ∈ V ∧ ( 𝐼 × 2o ) ∈ V ) → ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ∈ V ) |
30 |
24 28 29
|
sylancr |
⊢ ( 𝑋 ∈ 𝑊 → ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ∈ V ) |
31 |
23 30
|
fexd |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ∈ V ) |
32 |
|
fveq2 |
⊢ ( 𝑢 = 𝑋 → ( ♯ ‘ 𝑢 ) = ( ♯ ‘ 𝑋 ) ) |
33 |
32
|
oveq2d |
⊢ ( 𝑢 = 𝑋 → ( 0 ... ( ♯ ‘ 𝑢 ) ) = ( 0 ... ( ♯ ‘ 𝑋 ) ) ) |
34 |
|
eqidd |
⊢ ( 𝑢 = 𝑋 → ( 𝐼 × 2o ) = ( 𝐼 × 2o ) ) |
35 |
|
oveq1 |
⊢ ( 𝑢 = 𝑋 → ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) = ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) |
36 |
33 34 35
|
mpoeq123dv |
⊢ ( 𝑢 = 𝑋 → ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
37 |
|
oteq1 |
⊢ ( 𝑛 = 𝑎 → 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 = 〈 𝑎 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) |
38 |
|
oteq2 |
⊢ ( 𝑛 = 𝑎 → 〈 𝑎 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 = 〈 𝑎 , 𝑎 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) |
39 |
37 38
|
eqtrd |
⊢ ( 𝑛 = 𝑎 → 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 = 〈 𝑎 , 𝑎 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) |
40 |
39
|
oveq2d |
⊢ ( 𝑛 = 𝑎 → ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) = ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) |
41 |
|
id |
⊢ ( 𝑤 = 𝑏 → 𝑤 = 𝑏 ) |
42 |
|
fveq2 |
⊢ ( 𝑤 = 𝑏 → ( 𝑀 ‘ 𝑤 ) = ( 𝑀 ‘ 𝑏 ) ) |
43 |
41 42
|
s2eqd |
⊢ ( 𝑤 = 𝑏 → 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 = 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 ) |
44 |
43
|
oteq3d |
⊢ ( 𝑤 = 𝑏 → 〈 𝑎 , 𝑎 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 = 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) |
45 |
44
|
oveq2d |
⊢ ( 𝑤 = 𝑏 → ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) = ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) |
46 |
40 45
|
cbvmpov |
⊢ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) |
47 |
|
fveq2 |
⊢ ( 𝑣 = 𝑢 → ( ♯ ‘ 𝑣 ) = ( ♯ ‘ 𝑢 ) ) |
48 |
47
|
oveq2d |
⊢ ( 𝑣 = 𝑢 → ( 0 ... ( ♯ ‘ 𝑣 ) ) = ( 0 ... ( ♯ ‘ 𝑢 ) ) ) |
49 |
|
eqidd |
⊢ ( 𝑣 = 𝑢 → ( 𝐼 × 2o ) = ( 𝐼 × 2o ) ) |
50 |
|
oveq1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) = ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) |
51 |
48 49 50
|
mpoeq123dv |
⊢ ( 𝑣 = 𝑢 → ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
52 |
46 51
|
syl5eq |
⊢ ( 𝑣 = 𝑢 → ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
53 |
52
|
cbvmptv |
⊢ ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) ) = ( 𝑢 ∈ 𝑊 ↦ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
54 |
4 53
|
eqtri |
⊢ 𝑇 = ( 𝑢 ∈ 𝑊 ↦ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑢 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
55 |
36 54
|
fvmptg |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ∈ V ) → ( 𝑇 ‘ 𝑋 ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
56 |
31 55
|
mpdan |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝑇 ‘ 𝑋 ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ) |
57 |
56
|
feq1d |
⊢ ( 𝑋 ∈ 𝑊 → ( ( 𝑇 ‘ 𝑋 ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ↔ ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ) ) |
58 |
23 57
|
mpbird |
⊢ ( 𝑋 ∈ 𝑊 → ( 𝑇 ‘ 𝑋 ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ) |
59 |
56 58
|
jca |
⊢ ( 𝑋 ∈ 𝑊 → ( ( 𝑇 ‘ 𝑋 ) = ( 𝑎 ∈ ( 0 ... ( ♯ ‘ 𝑋 ) ) , 𝑏 ∈ ( 𝐼 × 2o ) ↦ ( 𝑋 splice 〈 𝑎 , 𝑎 , 〈“ 𝑏 ( 𝑀 ‘ 𝑏 ) ”〉 〉 ) ) ∧ ( 𝑇 ‘ 𝑋 ) : ( ( 0 ... ( ♯ ‘ 𝑋 ) ) × ( 𝐼 × 2o ) ) ⟶ 𝑊 ) ) |