Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbasmpt.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
6 |
|
prdsplusgval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
prdsplusgval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
8 |
|
prdsplusgval.p |
⊢ + = ( +g ‘ 𝑌 ) |
9 |
|
prdsplusgfval.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝐼 ) |
10 |
1 2 3 4 5 6 7 8
|
prdsplusgval |
⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
11 |
10
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 + 𝐺 ) ‘ 𝐽 ) = ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ‘ 𝐽 ) ) |
12 |
|
2fveq3 |
⊢ ( 𝑥 = 𝐽 → ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) = ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ) |
13 |
|
fveq2 |
⊢ ( 𝑥 = 𝐽 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐽 ) ) |
14 |
|
fveq2 |
⊢ ( 𝑥 = 𝐽 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝐽 ) ) |
15 |
12 13 14
|
oveq123d |
⊢ ( 𝑥 = 𝐽 → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝐽 ) ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ( 𝐺 ‘ 𝐽 ) ) ) |
16 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
17 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝐽 ) ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ( 𝐺 ‘ 𝐽 ) ) ∈ V |
18 |
15 16 17
|
fvmpt |
⊢ ( 𝐽 ∈ 𝐼 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ‘ 𝐽 ) = ( ( 𝐹 ‘ 𝐽 ) ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ( 𝐺 ‘ 𝐽 ) ) ) |
19 |
9 18
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ‘ 𝐽 ) = ( ( 𝐹 ‘ 𝐽 ) ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ( 𝐺 ‘ 𝐽 ) ) ) |
20 |
11 19
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐹 + 𝐺 ) ‘ 𝐽 ) = ( ( 𝐹 ‘ 𝐽 ) ( +g ‘ ( 𝑅 ‘ 𝐽 ) ) ( 𝐺 ‘ 𝐽 ) ) ) |