Step |
Hyp |
Ref |
Expression |
1 |
|
prjspnvs.e |
⊢ ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ 𝑆 𝑥 = ( 𝑙 · 𝑦 ) ) } |
2 |
|
prjspnvs.w |
⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ... 𝑁 ) ) |
3 |
|
prjspnvs.b |
⊢ 𝐵 = ( ( Base ‘ 𝑊 ) ∖ { ( 0g ‘ 𝑊 ) } ) |
4 |
|
prjspnvs.s |
⊢ 𝑆 = ( Base ‘ 𝐾 ) |
5 |
|
prjspnvs.x |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
6 |
|
prjspnvs.0 |
⊢ 0 = ( 0g ‘ 𝐾 ) |
7 |
|
prjspnvs.k |
⊢ ( 𝜑 → 𝐾 ∈ DivRing ) |
8 |
|
prjspnvs.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
9 |
|
prjspnvs.2 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑆 ) |
10 |
|
prjspnvs.3 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
11 |
|
ovexd |
⊢ ( 𝜑 → ( 0 ... 𝑁 ) ∈ V ) |
12 |
2
|
frlmlvec |
⊢ ( ( 𝐾 ∈ DivRing ∧ ( 0 ... 𝑁 ) ∈ V ) → 𝑊 ∈ LVec ) |
13 |
7 11 12
|
syl2anc |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
14 |
|
nelsn |
⊢ ( 𝐶 ≠ 0 → ¬ 𝐶 ∈ { 0 } ) |
15 |
10 14
|
syl |
⊢ ( 𝜑 → ¬ 𝐶 ∈ { 0 } ) |
16 |
9 15
|
eldifd |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝑆 ∖ { 0 } ) ) |
17 |
2
|
frlmsca |
⊢ ( ( 𝐾 ∈ DivRing ∧ ( 0 ... 𝑁 ) ∈ V ) → 𝐾 = ( Scalar ‘ 𝑊 ) ) |
18 |
7 11 17
|
syl2anc |
⊢ ( 𝜑 → 𝐾 = ( Scalar ‘ 𝑊 ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
20 |
4 19
|
syl5eq |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
21 |
18
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
22 |
6 21
|
syl5eq |
⊢ ( 𝜑 → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
23 |
22
|
sneqd |
⊢ ( 𝜑 → { 0 } = { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) |
24 |
20 23
|
difeq12d |
⊢ ( 𝜑 → ( 𝑆 ∖ { 0 } ) = ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) |
25 |
16 24
|
eleqtrd |
⊢ ( 𝜑 → 𝐶 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) |
26 |
|
eqid |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } |
27 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
28 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
29 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
30 |
26 3 27 5 28 29
|
prjspvs |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝐶 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( 𝐶 · 𝑋 ) { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } 𝑋 ) |
31 |
13 8 25 30
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 · 𝑋 ) { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } 𝑋 ) |
32 |
1 2 3 4 5
|
prjspnerlem |
⊢ ( 𝐾 ∈ DivRing → ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } ) |
33 |
7 32
|
syl |
⊢ ( 𝜑 → ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } ) |
34 |
33
|
breqd |
⊢ ( 𝜑 → ( ( 𝐶 · 𝑋 ) ∼ 𝑋 ↔ ( 𝐶 · 𝑋 ) { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝑥 = ( 𝑙 · 𝑦 ) ) } 𝑋 ) ) |
35 |
31 34
|
mpbird |
⊢ ( 𝜑 → ( 𝐶 · 𝑋 ) ∼ 𝑋 ) |