Step |
Hyp |
Ref |
Expression |
1 |
|
prdsinvgd2.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsinvgd2.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
3 |
|
prdsinvgd2.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsinvgd2.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Grp ) |
5 |
|
prdsinvgd2.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
6 |
|
prdsinvgd2.n |
⊢ 𝑁 = ( invg ‘ 𝑌 ) |
7 |
|
prdsinvgd2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
|
prdsinvgd2.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝐼 ) |
9 |
1 2 3 4 5 6 7
|
prdsinvgd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) ‘ 𝐽 ) = ( ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) ‘ 𝐽 ) ) |
11 |
|
2fveq3 |
⊢ ( 𝑥 = 𝐽 → ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) = ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 𝐽 → ( 𝑋 ‘ 𝑥 ) = ( 𝑋 ‘ 𝐽 ) ) |
13 |
11 12
|
fveq12d |
⊢ ( 𝑥 = 𝐽 → ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) = ( ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ‘ ( 𝑋 ‘ 𝐽 ) ) ) |
14 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) |
15 |
|
fvex |
⊢ ( ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ‘ ( 𝑋 ‘ 𝐽 ) ) ∈ V |
16 |
13 14 15
|
fvmpt |
⊢ ( 𝐽 ∈ 𝐼 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) ‘ 𝐽 ) = ( ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ‘ ( 𝑋 ‘ 𝐽 ) ) ) |
17 |
8 16
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑥 ) ) ‘ ( 𝑋 ‘ 𝑥 ) ) ) ‘ 𝐽 ) = ( ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ‘ ( 𝑋 ‘ 𝐽 ) ) ) |
18 |
10 17
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) ‘ 𝐽 ) = ( ( invg ‘ ( 𝑅 ‘ 𝐽 ) ) ‘ ( 𝑋 ‘ 𝐽 ) ) ) |