Step |
Hyp |
Ref |
Expression |
1 |
|
mexval.k |
⊢ 𝐾 = ( mTC ‘ 𝑇 ) |
2 |
|
mexval.e |
⊢ 𝐸 = ( mEx ‘ 𝑇 ) |
3 |
|
mexval2.c |
⊢ 𝐶 = ( mCN ‘ 𝑇 ) |
4 |
|
mexval2.v |
⊢ 𝑉 = ( mVR ‘ 𝑇 ) |
5 |
|
eqid |
⊢ ( mREx ‘ 𝑇 ) = ( mREx ‘ 𝑇 ) |
6 |
1 2 5
|
mexval |
⊢ 𝐸 = ( 𝐾 × ( mREx ‘ 𝑇 ) ) |
7 |
3 4 5
|
mrexval |
⊢ ( 𝑇 ∈ V → ( mREx ‘ 𝑇 ) = Word ( 𝐶 ∪ 𝑉 ) ) |
8 |
7
|
xpeq2d |
⊢ ( 𝑇 ∈ V → ( 𝐾 × ( mREx ‘ 𝑇 ) ) = ( 𝐾 × Word ( 𝐶 ∪ 𝑉 ) ) ) |
9 |
6 8
|
syl5eq |
⊢ ( 𝑇 ∈ V → 𝐸 = ( 𝐾 × Word ( 𝐶 ∪ 𝑉 ) ) ) |
10 |
|
0xp |
⊢ ( ∅ × Word ( 𝐶 ∪ 𝑉 ) ) = ∅ |
11 |
10
|
eqcomi |
⊢ ∅ = ( ∅ × Word ( 𝐶 ∪ 𝑉 ) ) |
12 |
|
fvprc |
⊢ ( ¬ 𝑇 ∈ V → ( mEx ‘ 𝑇 ) = ∅ ) |
13 |
2 12
|
syl5eq |
⊢ ( ¬ 𝑇 ∈ V → 𝐸 = ∅ ) |
14 |
|
fvprc |
⊢ ( ¬ 𝑇 ∈ V → ( mTC ‘ 𝑇 ) = ∅ ) |
15 |
1 14
|
syl5eq |
⊢ ( ¬ 𝑇 ∈ V → 𝐾 = ∅ ) |
16 |
15
|
xpeq1d |
⊢ ( ¬ 𝑇 ∈ V → ( 𝐾 × Word ( 𝐶 ∪ 𝑉 ) ) = ( ∅ × Word ( 𝐶 ∪ 𝑉 ) ) ) |
17 |
11 13 16
|
3eqtr4a |
⊢ ( ¬ 𝑇 ∈ V → 𝐸 = ( 𝐾 × Word ( 𝐶 ∪ 𝑉 ) ) ) |
18 |
9 17
|
pm2.61i |
⊢ 𝐸 = ( 𝐾 × Word ( 𝐶 ∪ 𝑉 ) ) |