Step |
Hyp |
Ref |
Expression |
1 |
|
qusaddf.u |
⊢ ( 𝜑 → 𝑈 = ( 𝑅 /s ∼ ) ) |
2 |
|
qusaddf.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
qusaddf.r |
⊢ ( 𝜑 → ∼ Er 𝑉 ) |
4 |
|
qusaddf.z |
⊢ ( 𝜑 → 𝑅 ∈ 𝑍 ) |
5 |
|
qusaddf.e |
⊢ ( 𝜑 → ( ( 𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞 ) → ( 𝑎 · 𝑏 ) ∼ ( 𝑝 · 𝑞 ) ) ) |
6 |
|
qusaddf.c |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( 𝑝 · 𝑞 ) ∈ 𝑉 ) |
7 |
|
qusaddflem.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ∼ ) |
8 |
|
qusaddflem.g |
⊢ ( 𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { 〈 〈 ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) 〉 , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) 〉 } ) |
9 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
10 |
2 9
|
eqeltrdi |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
11 |
|
erex |
⊢ ( ∼ Er 𝑉 → ( 𝑉 ∈ V → ∼ ∈ V ) ) |
12 |
3 10 11
|
sylc |
⊢ ( 𝜑 → ∼ ∈ V ) |
13 |
1 2 7 12 4
|
quslem |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ ( 𝑉 / ∼ ) ) |
14 |
3 10 7 6 5
|
ercpbl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 · 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ) |
15 |
13 14 8 6
|
imasaddflem |
⊢ ( 𝜑 → ∙ : ( ( 𝑉 / ∼ ) × ( 𝑉 / ∼ ) ) ⟶ ( 𝑉 / ∼ ) ) |