Step |
Hyp |
Ref |
Expression |
1 |
|
frgpval.m |
⊢ 𝐺 = ( freeGrp ‘ 𝐼 ) |
2 |
|
frgpval.b |
⊢ 𝑀 = ( freeMnd ‘ ( 𝐼 × 2o ) ) |
3 |
|
frgpval.r |
⊢ ∼ = ( ~FG ‘ 𝐼 ) |
4 |
|
elex |
⊢ ( 𝐼 ∈ 𝑉 → 𝐼 ∈ V ) |
5 |
|
xpeq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 × 2o ) = ( 𝐼 × 2o ) ) |
6 |
5
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( freeMnd ‘ ( 𝑖 × 2o ) ) = ( freeMnd ‘ ( 𝐼 × 2o ) ) ) |
7 |
6 2
|
eqtr4di |
⊢ ( 𝑖 = 𝐼 → ( freeMnd ‘ ( 𝑖 × 2o ) ) = 𝑀 ) |
8 |
|
fveq2 |
⊢ ( 𝑖 = 𝐼 → ( ~FG ‘ 𝑖 ) = ( ~FG ‘ 𝐼 ) ) |
9 |
8 3
|
eqtr4di |
⊢ ( 𝑖 = 𝐼 → ( ~FG ‘ 𝑖 ) = ∼ ) |
10 |
7 9
|
oveq12d |
⊢ ( 𝑖 = 𝐼 → ( ( freeMnd ‘ ( 𝑖 × 2o ) ) /s ( ~FG ‘ 𝑖 ) ) = ( 𝑀 /s ∼ ) ) |
11 |
|
df-frgp |
⊢ freeGrp = ( 𝑖 ∈ V ↦ ( ( freeMnd ‘ ( 𝑖 × 2o ) ) /s ( ~FG ‘ 𝑖 ) ) ) |
12 |
|
ovex |
⊢ ( 𝑀 /s ∼ ) ∈ V |
13 |
10 11 12
|
fvmpt |
⊢ ( 𝐼 ∈ V → ( freeGrp ‘ 𝐼 ) = ( 𝑀 /s ∼ ) ) |
14 |
4 13
|
syl |
⊢ ( 𝐼 ∈ 𝑉 → ( freeGrp ‘ 𝐼 ) = ( 𝑀 /s ∼ ) ) |
15 |
1 14
|
syl5eq |
⊢ ( 𝐼 ∈ 𝑉 → 𝐺 = ( 𝑀 /s ∼ ) ) |