Step |
Hyp |
Ref |
Expression |
1 |
|
frmdval.m |
⊢ 𝑀 = ( freeMnd ‘ 𝐼 ) |
2 |
|
frmdval.b |
⊢ ( 𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼 ) |
3 |
|
frmdval.p |
⊢ + = ( ++ ↾ ( 𝐵 × 𝐵 ) ) |
4 |
|
df-frmd |
⊢ freeMnd = ( 𝑖 ∈ V ↦ { 〈 ( Base ‘ ndx ) , Word 𝑖 〉 , 〈 ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) 〉 } ) |
5 |
|
wrdeq |
⊢ ( 𝑖 = 𝐼 → Word 𝑖 = Word 𝐼 ) |
6 |
2
|
eqcomd |
⊢ ( 𝐼 ∈ 𝑉 → Word 𝐼 = 𝐵 ) |
7 |
5 6
|
sylan9eqr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → Word 𝑖 = 𝐵 ) |
8 |
7
|
opeq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → 〈 ( Base ‘ ndx ) , Word 𝑖 〉 = 〈 ( Base ‘ ndx ) , 𝐵 〉 ) |
9 |
7
|
sqxpeqd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( Word 𝑖 × Word 𝑖 ) = ( 𝐵 × 𝐵 ) ) |
10 |
9
|
reseq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) = ( ++ ↾ ( 𝐵 × 𝐵 ) ) ) |
11 |
10 3
|
eqtr4di |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) = + ) |
12 |
11
|
opeq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → 〈 ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) 〉 = 〈 ( +g ‘ ndx ) , + 〉 ) |
13 |
8 12
|
preq12d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → { 〈 ( Base ‘ ndx ) , Word 𝑖 〉 , 〈 ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 } ) |
14 |
|
elex |
⊢ ( 𝐼 ∈ 𝑉 → 𝐼 ∈ V ) |
15 |
|
prex |
⊢ { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 } ∈ V |
16 |
15
|
a1i |
⊢ ( 𝐼 ∈ 𝑉 → { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 } ∈ V ) |
17 |
4 13 14 16
|
fvmptd2 |
⊢ ( 𝐼 ∈ 𝑉 → ( freeMnd ‘ 𝐼 ) = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 } ) |
18 |
1 17
|
syl5eq |
⊢ ( 𝐼 ∈ 𝑉 → 𝑀 = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 } ) |