Step |
Hyp |
Ref |
Expression |
1 |
|
prjspnssbas.p |
⊢ 𝑃 = ( 𝑁 ℙ𝕣𝕠𝕛n 𝐾 ) |
2 |
|
prjspnssbas.w |
⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ... 𝑁 ) ) |
3 |
|
prjspnssbas.b |
⊢ 𝐵 = ( ( Base ‘ 𝑊 ) ∖ { ( 0g ‘ 𝑊 ) } ) |
4 |
|
prjspnssbas.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
5 |
|
prjspnssbas.k |
⊢ ( 𝜑 → 𝐾 ∈ DivRing ) |
6 |
|
eqid |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
9 |
6 2 3 7 8
|
prjspnval2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing ) → ( 𝑁 ℙ𝕣𝕠𝕛n 𝐾 ) = ( 𝐵 / { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } ) ) |
10 |
4 5 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ℙ𝕣𝕠𝕛n 𝐾 ) = ( 𝐵 / { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } ) ) |
11 |
1 10
|
eqtrid |
⊢ ( 𝜑 → 𝑃 = ( 𝐵 / { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } ) ) |
12 |
6 2 3 7 8 5
|
prjspner |
⊢ ( 𝜑 → { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } Er 𝐵 ) |
13 |
12
|
qsss |
⊢ ( 𝜑 → ( 𝐵 / { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ 𝐾 ) 𝑥 = ( 𝑙 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) } ) ⊆ 𝒫 𝐵 ) |
14 |
11 13
|
eqsstrd |
⊢ ( 𝜑 → 𝑃 ⊆ 𝒫 𝐵 ) |