Step |
Hyp |
Ref |
Expression |
1 |
|
mamufval.f |
⊢ 𝐹 = ( 𝑅 maMul 〈 𝑀 , 𝑁 , 𝑃 〉 ) |
2 |
|
mamufval.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
mamufval.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
mamufval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
5 |
|
mamufval.m |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
6 |
|
mamufval.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
7 |
|
mamufval.p |
⊢ ( 𝜑 → 𝑃 ∈ Fin ) |
8 |
|
mamuval.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
9 |
|
mamuval.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑃 ) ) ) |
10 |
1 2 3 4 5 6 7
|
mamufval |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) , 𝑦 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑃 ) ) ↦ ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) ) ) ) |
11 |
|
oveq |
⊢ ( 𝑥 = 𝑋 → ( 𝑖 𝑥 𝑗 ) = ( 𝑖 𝑋 𝑗 ) ) |
12 |
|
oveq |
⊢ ( 𝑦 = 𝑌 → ( 𝑗 𝑦 𝑘 ) = ( 𝑗 𝑌 𝑘 ) ) |
13 |
11 12
|
oveqan12d |
⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) = ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) = ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) |
15 |
14
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) |
16 |
15
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) |
17 |
16
|
mpoeq3dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ) |
18 |
|
mpoexga |
⊢ ( ( 𝑀 ∈ Fin ∧ 𝑃 ∈ Fin ) → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ∈ V ) |
19 |
5 7 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ∈ V ) |
20 |
10 17 8 9 19
|
ovmpod |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ) |