Step |
Hyp |
Ref |
Expression |
1 |
|
selvffval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
2 |
|
selvffval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
3 |
|
df-selv |
⊢ selectVars = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) ) ) ) |
4 |
3
|
a1i |
⊢ ( 𝜑 → selectVars = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) ) ) ) ) |
5 |
|
pweq |
⊢ ( 𝑖 = 𝐼 → 𝒫 𝑖 = 𝒫 𝐼 ) |
6 |
5
|
adantr |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → 𝒫 𝑖 = 𝒫 𝐼 ) |
7 |
|
oveq12 |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑖 mPoly 𝑟 ) = ( 𝐼 mPoly 𝑅 ) ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ) |
9 |
|
difeq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 ∖ 𝑗 ) = ( 𝐼 ∖ 𝑗 ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑖 ∖ 𝑗 ) = ( 𝐼 ∖ 𝑗 ) ) |
11 |
|
simpr |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → 𝑟 = 𝑅 ) |
12 |
10 11
|
oveq12d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) = ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) ) |
13 |
|
oveq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 evalSub 𝑡 ) = ( 𝐼 evalSub 𝑡 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑖 evalSub 𝑡 ) = ( 𝐼 evalSub 𝑡 ) ) |
15 |
14
|
fveq1d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) = ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ) |
16 |
15
|
fveq1d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) = ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ) |
17 |
|
simpl |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → 𝑖 = 𝐼 ) |
18 |
10 11
|
oveq12d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) = ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ) |
19 |
18
|
fveq1d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) = ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) = ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) |
21 |
20
|
ifeq2d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) = if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) |
22 |
17 21
|
mpteq12dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) |
23 |
16 22
|
fveq12d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) = ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
24 |
23
|
csbeq2dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) = ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
25 |
24
|
csbeq2dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) = ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
26 |
25
|
csbeq2dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) = ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
27 |
12 26
|
csbeq12dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) = ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
28 |
8 27
|
mpteq12dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) ) = ( 𝑓 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ↦ ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) ) |
29 |
6 28
|
mpteq12dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) ) ) = ( 𝑗 ∈ 𝒫 𝐼 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ↦ ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) ) ) |
30 |
29
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) ) → ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ ⦋ ( ( 𝑖 ∖ 𝑗 ) mPoly 𝑟 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝑖 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝑖 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝑖 ∖ 𝑗 ) mVar 𝑟 ) ‘ 𝑥 ) ) ) ) ) ) ) = ( 𝑗 ∈ 𝒫 𝐼 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ↦ ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) ) ) |
31 |
1
|
elexd |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
32 |
2
|
elexd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
33 |
1
|
pwexd |
⊢ ( 𝜑 → 𝒫 𝐼 ∈ V ) |
34 |
33
|
mptexd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝒫 𝐼 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ↦ ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) ) ∈ V ) |
35 |
4 30 31 32 34
|
ovmpod |
⊢ ( 𝜑 → ( 𝐼 selectVars 𝑅 ) = ( 𝑗 ∈ 𝒫 𝐼 ↦ ( 𝑓 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ↦ ⦋ ( ( 𝐼 ∖ 𝑗 ) mPoly 𝑅 ) / 𝑢 ⦌ ⦋ ( 𝑗 mPoly 𝑢 ) / 𝑡 ⦌ ⦋ ( algSc ‘ 𝑡 ) / 𝑐 ⦌ ⦋ ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) / 𝑑 ⦌ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝑗 , ( ( 𝑗 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝑗 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) ) ) |