Step |
Hyp |
Ref |
Expression |
1 |
|
resvsca.r |
⊢ 𝑅 = ( 𝑊 ↾v 𝐴 ) |
2 |
|
resvsca.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
resvsca.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
4 |
|
elex |
⊢ ( 𝑊 ∈ 𝑋 → 𝑊 ∈ V ) |
5 |
|
elex |
⊢ ( 𝐴 ∈ 𝑌 → 𝐴 ∈ V ) |
6 |
|
ovex |
⊢ ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ∈ V |
7 |
|
ifcl |
⊢ ( ( 𝑊 ∈ V ∧ ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ∈ V ) → if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ∈ V ) |
8 |
6 7
|
mpan2 |
⊢ ( 𝑊 ∈ V → if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ∈ V ) |
9 |
8
|
adantr |
⊢ ( ( 𝑊 ∈ V ∧ 𝐴 ∈ V ) → if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ∈ V ) |
10 |
|
simpl |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → 𝑤 = 𝑊 ) |
11 |
10
|
fveq2d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( Scalar ‘ 𝑤 ) = ( Scalar ‘ 𝑊 ) ) |
12 |
11 2
|
eqtr4di |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( Scalar ‘ 𝑤 ) = 𝐹 ) |
13 |
12
|
fveq2d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( Base ‘ ( Scalar ‘ 𝑤 ) ) = ( Base ‘ 𝐹 ) ) |
14 |
13 3
|
eqtr4di |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( Base ‘ ( Scalar ‘ 𝑤 ) ) = 𝐵 ) |
15 |
|
simpr |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → 𝑥 = 𝐴 ) |
16 |
14 15
|
sseq12d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( ( Base ‘ ( Scalar ‘ 𝑤 ) ) ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴 ) ) |
17 |
12 15
|
oveq12d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( ( Scalar ‘ 𝑤 ) ↾s 𝑥 ) = ( 𝐹 ↾s 𝐴 ) ) |
18 |
17
|
opeq2d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑤 ) ↾s 𝑥 ) 〉 = 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) |
19 |
10 18
|
oveq12d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → ( 𝑤 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑤 ) ↾s 𝑥 ) 〉 ) = ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) |
20 |
16 10 19
|
ifbieq12d |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑥 = 𝐴 ) → if ( ( Base ‘ ( Scalar ‘ 𝑤 ) ) ⊆ 𝑥 , 𝑤 , ( 𝑤 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑤 ) ↾s 𝑥 ) 〉 ) ) = if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ) |
21 |
|
df-resv |
⊢ ↾v = ( 𝑤 ∈ V , 𝑥 ∈ V ↦ if ( ( Base ‘ ( Scalar ‘ 𝑤 ) ) ⊆ 𝑥 , 𝑤 , ( 𝑤 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑤 ) ↾s 𝑥 ) 〉 ) ) ) |
22 |
20 21
|
ovmpoga |
⊢ ( ( 𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ∈ V ) → ( 𝑊 ↾v 𝐴 ) = if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ) |
23 |
9 22
|
mpd3an3 |
⊢ ( ( 𝑊 ∈ V ∧ 𝐴 ∈ V ) → ( 𝑊 ↾v 𝐴 ) = if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ) |
24 |
4 5 23
|
syl2an |
⊢ ( ( 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌 ) → ( 𝑊 ↾v 𝐴 ) = if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ) |
25 |
1 24
|
syl5eq |
⊢ ( ( 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌 ) → 𝑅 = if ( 𝐵 ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( 𝐹 ↾s 𝐴 ) 〉 ) ) ) |