| Step |
Hyp |
Ref |
Expression |
| 1 |
|
selvval.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
| 2 |
|
selvval.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
| 3 |
|
selvval.u |
⊢ 𝑈 = ( ( 𝐼 ∖ 𝐽 ) mPoly 𝑅 ) |
| 4 |
|
selvval.t |
⊢ 𝑇 = ( 𝐽 mPoly 𝑈 ) |
| 5 |
|
selvval.c |
⊢ 𝐶 = ( algSc ‘ 𝑇 ) |
| 6 |
|
selvval.d |
⊢ 𝐷 = ( 𝐶 ∘ ( algSc ‘ 𝑈 ) ) |
| 7 |
|
selvval.j |
⊢ ( 𝜑 → 𝐽 ⊆ 𝐼 ) |
| 8 |
|
selvval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
| 9 |
|
coeq2 |
⊢ ( 𝑓 = 𝐹 → ( 𝑑 ∘ 𝑓 ) = ( 𝑑 ∘ 𝐹 ) ) |
| 10 |
9
|
fveq2d |
⊢ ( 𝑓 = 𝐹 → ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝑓 ) ) = ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝐹 ) ) ) |
| 11 |
10
|
fveq1d |
⊢ ( 𝑓 = 𝐹 → ( ( ( ( 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 ) ) ) ) ) ) |
| 12 |
11
|
csbeq2dv |
⊢ ( 𝑓 = 𝐹 → ⦋ ( 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 ) ) ) ) ) ) |
| 13 |
12
|
csbeq2dv |
⊢ ( 𝑓 = 𝐹 → ⦋ ( 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 ) ) ) ) ) ) |
| 14 |
13
|
csbeq2dv |
⊢ ( 𝑓 = 𝐹 → ⦋ ( 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 ) ) ) ) ) ) |
| 15 |
14
|
csbeq2dv |
⊢ ( 𝑓 = 𝐹 → ⦋ ( ( 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 ) ) ) ) ) ) |
| 16 |
|
reldmmpl |
⊢ Rel dom mPoly |
| 17 |
16 1 2
|
elbasov |
⊢ ( 𝐹 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
| 18 |
8 17
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
| 19 |
18
|
simpld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
| 20 |
18
|
simprd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
| 21 |
19 20 7
|
selvfval |
⊢ ( 𝜑 → ( ( 𝐼 selectVars 𝑅 ) ‘ 𝐽 ) = ( 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 ) ) ) ) ) ) ) |
| 22 |
1
|
fveq2i |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) |
| 23 |
2 22
|
eqtri |
⊢ 𝐵 = ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) |
| 24 |
8 23
|
eleqtrdi |
⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ) |
| 25 |
|
fvex |
⊢ ( ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑢 ) ‘ 𝑥 ) , ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ∈ V |
| 26 |
25
|
csbex |
⊢ ⦋ ( 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 |
| 27 |
26
|
csbex |
⊢ ⦋ ( 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 |
| 28 |
27
|
csbex |
⊢ ⦋ ( 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 |
| 29 |
28
|
csbex |
⊢ ⦋ ( ( 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 |
| 30 |
29
|
a1i |
⊢ ( 𝜑 → ⦋ ( ( 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 ) |
| 31 |
15 21 24 30
|
fvmptd4 |
⊢ ( 𝜑 → ( ( ( 𝐼 selectVars 𝑅 ) ‘ 𝐽 ) ‘ 𝐹 ) = ⦋ ( ( 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 ) ) ) ) ) ) |
| 32 |
|
ovex |
⊢ ( ( 𝐼 ∖ 𝐽 ) mPoly 𝑅 ) ∈ V |
| 33 |
3
|
eqeq2i |
⊢ ( 𝑢 = 𝑈 ↔ 𝑢 = ( ( 𝐼 ∖ 𝐽 ) mPoly 𝑅 ) ) |
| 34 |
|
oveq2 |
⊢ ( 𝑢 = 𝑈 → ( 𝐽 mPoly 𝑢 ) = ( 𝐽 mPoly 𝑈 ) ) |
| 35 |
|
fveq2 |
⊢ ( 𝑢 = 𝑈 → ( algSc ‘ 𝑢 ) = ( algSc ‘ 𝑈 ) ) |
| 36 |
35
|
coeq2d |
⊢ ( 𝑢 = 𝑈 → ( 𝑐 ∘ ( algSc ‘ 𝑢 ) ) = ( 𝑐 ∘ ( algSc ‘ 𝑈 ) ) ) |
| 37 |
|
oveq2 |
⊢ ( 𝑢 = 𝑈 → ( 𝐽 mVar 𝑢 ) = ( 𝐽 mVar 𝑈 ) ) |
| 38 |
37
|
fveq1d |
⊢ ( 𝑢 = 𝑈 → ( ( 𝐽 mVar 𝑢 ) ‘ 𝑥 ) = ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) ) |
| 39 |
38
|
ifeq1d |
⊢ ( 𝑢 = 𝑈 → 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 ) ) ) ) |
| 40 |
39
|
mpteq2dv |
⊢ ( 𝑢 = 𝑈 → ( 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 ) ) ) ) ) |
| 41 |
40
|
fveq2d |
⊢ ( 𝑢 = 𝑈 → ( ( ( ( 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 ) ) ) ) ) ) |
| 42 |
36 41
|
csbeq12dv |
⊢ ( 𝑢 = 𝑈 → ⦋ ( 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 ) ) ) ) ) ) |
| 43 |
42
|
csbeq2dv |
⊢ ( 𝑢 = 𝑈 → ⦋ ( 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 ) ) ) ) ) ) |
| 44 |
34 43
|
csbeq12dv |
⊢ ( 𝑢 = 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 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 ) ) ) ) ) ) |
| 45 |
|
ovex |
⊢ ( 𝐽 mPoly 𝑈 ) ∈ V |
| 46 |
4
|
eqeq2i |
⊢ ( 𝑡 = 𝑇 ↔ 𝑡 = ( 𝐽 mPoly 𝑈 ) ) |
| 47 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( algSc ‘ 𝑡 ) = ( algSc ‘ 𝑇 ) ) |
| 48 |
|
oveq2 |
⊢ ( 𝑡 = 𝑇 → ( 𝐼 evalSub 𝑡 ) = ( 𝐼 evalSub 𝑇 ) ) |
| 49 |
48
|
fveq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) = ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝑑 ) ) |
| 50 |
49
|
fveq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( 𝐼 evalSub 𝑡 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝐹 ) ) = ( ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝐹 ) ) ) |
| 51 |
50
|
fveq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( ( 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 ) ) ) ) ) ) |
| 52 |
51
|
csbeq2dv |
⊢ ( 𝑡 = 𝑇 → ⦋ ( 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 ) ) ) ) ) ) |
| 53 |
47 52
|
csbeq12dv |
⊢ ( 𝑡 = 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 ) ) ) ) ) = ⦋ ( 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 ) ) ) ) ) ) |
| 54 |
|
fvex |
⊢ ( algSc ‘ 𝑇 ) ∈ V |
| 55 |
5
|
eqeq2i |
⊢ ( 𝑐 = 𝐶 ↔ 𝑐 = ( algSc ‘ 𝑇 ) ) |
| 56 |
|
coeq1 |
⊢ ( 𝑐 = 𝐶 → ( 𝑐 ∘ ( algSc ‘ 𝑈 ) ) = ( 𝐶 ∘ ( algSc ‘ 𝑈 ) ) ) |
| 57 |
|
fveq1 |
⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) = ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) |
| 58 |
57
|
ifeq2d |
⊢ ( 𝑐 = 𝐶 → 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 ) ) ) ) |
| 59 |
58
|
mpteq2dv |
⊢ ( 𝑐 = 𝐶 → ( 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 ) ) ) ) ) |
| 60 |
59
|
fveq2d |
⊢ ( 𝑐 = 𝐶 → ( ( ( ( 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 ) ) ) ) ) ) |
| 61 |
56 60
|
csbeq12dv |
⊢ ( 𝑐 = 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 ) ) ) ) ) = ⦋ ( 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 ) ) ) ) ) ) |
| 62 |
5
|
fvexi |
⊢ 𝐶 ∈ V |
| 63 |
|
fvex |
⊢ ( algSc ‘ 𝑈 ) ∈ V |
| 64 |
62 63
|
coex |
⊢ ( 𝐶 ∘ ( algSc ‘ 𝑈 ) ) ∈ V |
| 65 |
6
|
eqeq2i |
⊢ ( 𝑑 = 𝐷 ↔ 𝑑 = ( 𝐶 ∘ ( algSc ‘ 𝑈 ) ) ) |
| 66 |
|
rneq |
⊢ ( 𝑑 = 𝐷 → ran 𝑑 = ran 𝐷 ) |
| 67 |
66
|
fveq2d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝑑 ) = ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝐷 ) ) |
| 68 |
|
coeq1 |
⊢ ( 𝑑 = 𝐷 → ( 𝑑 ∘ 𝐹 ) = ( 𝐷 ∘ 𝐹 ) ) |
| 69 |
67 68
|
fveq12d |
⊢ ( 𝑑 = 𝐷 → ( ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝑑 ) ‘ ( 𝑑 ∘ 𝐹 ) ) = ( ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ) |
| 70 |
69
|
fveq1d |
⊢ ( 𝑑 = 𝐷 → ( ( ( ( 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 ) ) ) ) ) ) |
| 71 |
65 70
|
sylbir |
⊢ ( 𝑑 = ( 𝐶 ∘ ( algSc ‘ 𝑈 ) ) → ( ( ( ( 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 ) ) ) ) ) ) |
| 72 |
64 71
|
csbie |
⊢ ⦋ ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) |
| 73 |
61 72
|
eqtrdi |
⊢ ( 𝑐 = 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 74 |
55 73
|
sylbir |
⊢ ( 𝑐 = ( algSc ‘ 𝑇 ) → ⦋ ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 75 |
54 74
|
csbie |
⊢ ⦋ ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) |
| 76 |
53 75
|
eqtrdi |
⊢ ( 𝑡 = 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 77 |
46 76
|
sylbir |
⊢ ( 𝑡 = ( 𝐽 mPoly 𝑈 ) → ⦋ ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 78 |
45 77
|
csbie |
⊢ ⦋ ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) |
| 79 |
44 78
|
eqtrdi |
⊢ ( 𝑢 = 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 80 |
33 79
|
sylbir |
⊢ ( 𝑢 = ( ( 𝐼 ∖ 𝐽 ) mPoly 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 ) ) ) ) ) = ( ( ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |
| 81 |
32 80
|
csbie |
⊢ ⦋ ( ( 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 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) |
| 82 |
31 81
|
eqtrdi |
⊢ ( 𝜑 → ( ( ( 𝐼 selectVars 𝑅 ) ‘ 𝐽 ) ‘ 𝐹 ) = ( ( ( ( 𝐼 evalSub 𝑇 ) ‘ ran 𝐷 ) ‘ ( 𝐷 ∘ 𝐹 ) ) ‘ ( 𝑥 ∈ 𝐼 ↦ if ( 𝑥 ∈ 𝐽 , ( ( 𝐽 mVar 𝑈 ) ‘ 𝑥 ) , ( 𝐶 ‘ ( ( ( 𝐼 ∖ 𝐽 ) mVar 𝑅 ) ‘ 𝑥 ) ) ) ) ) ) |