Step | Hyp | Ref
| Expression |
1 | | selvvvval.p |
. . . . . 6
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
2 | | selvvvval.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑃) |
3 | | eqid 2724 |
. . . . . 6
⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅) |
4 | | eqid 2724 |
. . . . . 6
⊢ (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) |
5 | | eqid 2724 |
. . . . . 6
⊢
(algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
6 | | eqid 2724 |
. . . . . 6
⊢
((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
7 | | selvvvval.i |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
8 | | selvvvval.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ CRing) |
9 | | selvvvval.j |
. . . . . 6
⊢ (𝜑 → 𝐽 ⊆ 𝐼) |
10 | | selvvvval.f |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | selvval2 41645 |
. . . . 5
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = (((𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹))‘(𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))))) |
12 | | eqid 2724 |
. . . . . 6
⊢ (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
13 | | eqid 2724 |
. . . . . 6
⊢ (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
14 | | eqid 2724 |
. . . . . 6
⊢
(Base‘(𝐼 mPoly
(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
15 | | selvvvval.d |
. . . . . 6
⊢ 𝐷 = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
16 | | eqid 2724 |
. . . . . 6
⊢
(Base‘(𝐽 mPoly
((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
17 | | eqid 2724 |
. . . . . 6
⊢
(mulGrp‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
18 | | eqid 2724 |
. . . . . 6
⊢
(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) =
(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
19 | | eqid 2724 |
. . . . . 6
⊢
(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
20 | 7, 9 | ssexd 5314 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ V) |
21 | 7 | difexd 5319 |
. . . . . . . 8
⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) |
22 | 3, 21, 8 | mplcrngd 41607 |
. . . . . . 7
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing) |
23 | 4, 20, 22 | mplcrngd 41607 |
. . . . . 6
⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing) |
24 | 4 | mplassa 21891 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ V ∧ ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg) |
25 | 20, 22, 24 | syl2anc 583 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg) |
26 | | eqid 2724 |
. . . . . . . . . . 11
⊢
(Scalar‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
27 | 5, 26 | asclrhm 21752 |
. . . . . . . . . 10
⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
28 | 25, 27 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
29 | 3 | mplassa 21891 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg) |
30 | 21, 8, 29 | syl2anc 583 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg) |
31 | | eqid 2724 |
. . . . . . . . . . . 12
⊢
(algSc‘((𝐼
∖ 𝐽) mPoly 𝑅)) = (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
32 | | eqid 2724 |
. . . . . . . . . . . 12
⊢
(Scalar‘((𝐼
∖ 𝐽) mPoly 𝑅)) = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
33 | 31, 32 | asclrhm 21752 |
. . . . . . . . . . 11
⊢ (((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
34 | 30, 33 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
35 | 3, 21, 8 | mplsca 21882 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 = (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
36 | 35 | eqcomd 2730 |
. . . . . . . . . . 11
⊢ (𝜑 → (Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = 𝑅) |
37 | 4, 20, 22 | mplsca 21882 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
38 | 36, 37 | oveq12d 7419 |
. . . . . . . . . 10
⊢ (𝜑 → ((Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)) RingHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
39 | 34, 38 | eleqtrd 2827 |
. . . . . . . . 9
⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
40 | | rhmco 20393 |
. . . . . . . . 9
⊢
(((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∧ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ (𝑅 RingHom (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
41 | 28, 39, 40 | syl2anc 583 |
. . . . . . . 8
⊢ (𝜑 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
42 | | rhmghm 20376 |
. . . . . . . 8
⊢
(((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 GrpHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
43 | | ghmmhm 19141 |
. . . . . . . 8
⊢
(((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 GrpHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 MndHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
44 | 41, 42, 43 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (𝑅 MndHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
45 | 1, 13, 2, 14, 7, 44, 10 | mhmcompl 41609 |
. . . . . 6
⊢ (𝜑 → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹) ∈ (Base‘(𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
46 | | fvexd 6896 |
. . . . . . 7
⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V) |
47 | | eqid 2724 |
. . . . . . . . . . . 12
⊢ (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) = (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)) |
48 | 22 | crngringd 20141 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring) |
49 | 4, 47, 16, 20, 48 | mvrf2 21862 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅)):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
50 | 49 | ffvelcdmda 7076 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
51 | 50 | adantlr 712 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
52 | | eldif 3950 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝐼 ∖ 𝐽) ↔ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽)) |
53 | | eqid 2724 |
. . . . . . . . . . . . . 14
⊢
(Base‘((𝐼
∖ 𝐽) mPoly 𝑅)) = (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
54 | 4, 16, 53, 5, 20, 48 | mplasclf 21936 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
55 | 54 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
56 | | eqid 2724 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅) |
57 | 8 | crngringd 20141 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ Ring) |
58 | 3, 56, 53, 21, 57 | mvrf2 21862 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
59 | 58 | ffvelcdmda 7076 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
60 | 55, 59 | ffvelcdmd 7077 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
61 | 52, 60 | sylan2br 594 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐼 ∧ ¬ 𝑧 ∈ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
62 | 61 | anassrs 467 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ ¬ 𝑧 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
63 | 51, 62 | ifclda 4555 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
64 | 63 | fmpttd 7106 |
. . . . . . 7
⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))):𝐼⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
65 | 46, 7, 64 | elmapdd 8831 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))) ∈ ((Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↑m 𝐼)) |
66 | 12, 13, 14, 15, 16, 17, 18, 19, 7, 23, 45, 65 | evlvvval 41634 |
. . . . 5
⊢ (𝜑 → (((𝐼 eval (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹))‘(𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))))))) |
67 | | eqid 2724 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑅) =
(Base‘𝑅) |
68 | 1, 67, 2, 15, 10 | mplelf 21867 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑅)) |
69 | 68 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐹:𝐷⟶(Base‘𝑅)) |
70 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 ∈ 𝐷) |
71 | 69, 70 | fvco3d 6981 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝐹‘𝑔))) |
72 | 3, 53, 67, 31, 21, 57 | mplasclf 21936 |
. . . . . . . . . . . 12
⊢ (𝜑 → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)):(Base‘𝑅)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
73 | 72 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)):(Base‘𝑅)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
74 | 68 | ffvelcdmda 7076 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘𝑅)) |
75 | 73, 74 | fvco3d 6981 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝐹‘𝑔)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))) |
76 | 71, 75 | eqtrd 2764 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))) |
77 | 17, 16 | mgpbas 20035 |
. . . . . . . . . . . 12
⊢
(Base‘(𝐽 mPoly
((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
78 | | eqid 2724 |
. . . . . . . . . . . 12
⊢
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) =
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
79 | 17, 19 | mgpplusg 20033 |
. . . . . . . . . . . 12
⊢
(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) =
(+g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
80 | 17 | crngmgp 20136 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CRing → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd) |
81 | 23, 80 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd) |
82 | 81 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CMnd) |
83 | 7 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐼 ∈ 𝑉) |
84 | 81 | cmnmndd 19714 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd) |
85 | 84 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd) |
86 | 15 | psrbagf 21780 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 ∈ 𝐷 → 𝑔:𝐼⟶ℕ0) |
87 | 86 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔:𝐼⟶ℕ0) |
88 | 87 | ffvelcdmda 7076 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈
ℕ0) |
89 | | eqid 2724 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)))) |
90 | | eleq1w 2808 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝐽 ↔ 𝑘 ∈ 𝐽)) |
91 | | fveq2 6881 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑘 → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) |
92 | | fveq2 6881 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑘 → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) |
93 | 92 | fveq2d 6885 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑘 → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧)) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
94 | 90, 91, 93 | ifbieq12d 4548 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑘 → if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
95 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼) |
96 | 49 | ffvelcdmda 7076 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
97 | 96 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
98 | 97 | adantlr 712 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ 𝑘 ∈ 𝐽) → ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
99 | | eldif 3950 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) ↔ (𝑘 ∈ 𝐼 ∧ ¬ 𝑘 ∈ 𝐽)) |
100 | 54 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
101 | 58 | ffvelcdmda 7076 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
102 | 100, 101 | ffvelcdmd 7077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
103 | 99, 102 | sylan2br 594 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐼 ∧ ¬ 𝑘 ∈ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
104 | 103 | anassrs 467 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑘 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
105 | 104 | adantllr 716 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) ∧ ¬ 𝑘 ∈ 𝐽) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
106 | 98, 105 | ifclda 4555 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
107 | 89, 94, 95, 106 | fvmptd3 7011 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
108 | 107, 106 | eqeltrd 2825 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
109 | 77, 18, 85, 88, 108 | mulgnn0cld 19012 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐼) → ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
110 | 109 | fmpttd 7106 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))):𝐼⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
111 | 7 | mptexd 7217 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ∈ V) |
112 | 111 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ∈ V) |
113 | | fvexd 6896 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) →
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V) |
114 | | funmpt 6576 |
. . . . . . . . . . . . . 14
⊢ Fun
(𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) |
115 | 114 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → Fun (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) |
116 | 87 | feqmptd 6950 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 = (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘))) |
117 | 15 | psrbagfsupp 21782 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 ∈ 𝐷 → 𝑔 finSupp 0) |
118 | 117 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 finSupp 0) |
119 | 116, 118 | eqbrtrrd 5162 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) finSupp 0) |
120 | | ssidd 3997 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) supp 0) ⊆ ((𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) supp 0)) |
121 | 77, 78, 18 | mulg0 18992 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) →
(0(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))𝑡) = (0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
122 | 121 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑡 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) →
(0(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))𝑡) = (0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
123 | | 0zd 12567 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 0 ∈ ℤ) |
124 | 120, 122,
88, 108, 123 | suppssov1 8177 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) supp
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ⊆ ((𝑘 ∈ 𝐼 ↦ (𝑔‘𝑘)) supp 0)) |
125 | 112, 113,
115, 119, 124 | fsuppsssuppgd 41557 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) finSupp
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
126 | | disjdifr 4464 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅ |
127 | 126 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) |
128 | | undifr 4474 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
129 | 9, 128 | sylib 217 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
130 | 129 | eqcomd 2730 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 = ((𝐼 ∖ 𝐽) ∪ 𝐽)) |
131 | 130 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐼 = ((𝐼 ∖ 𝐽) ∪ 𝐽)) |
132 | 77, 78, 79, 82, 83, 110, 125, 127, 131 | gsumsplit 19838 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽)))) |
133 | | difssd 4124 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼) |
134 | 133 | resmptd 6030 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) |
135 | | eldifi 4118 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → 𝑘 ∈ 𝐼) |
136 | 135 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ 𝐼) |
137 | 135, 106 | sylan2 592 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
138 | 89, 94, 136, 137 | fvmptd3 7011 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
139 | | eldifn 4119 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (𝐼 ∖ 𝐽) → ¬ 𝑘 ∈ 𝐽) |
140 | 139 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ¬ 𝑘 ∈ 𝐽) |
141 | 140 | iffalsed 4531 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
142 | 138, 141 | eqtrd 2764 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
143 | 142 | oveq2d 7417 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
144 | 143 | mpteq2dva 5238 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
145 | 134, 144 | eqtrd 2764 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
146 | 145 | oveq2d 7417 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))) |
147 | 9 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐽 ⊆ 𝐼) |
148 | 147 | resmptd 6030 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) |
149 | 9 | sselda 3974 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐼) |
150 | 149 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐼) |
151 | 150, 106 | syldan 590 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
152 | 89, 94, 150, 151 | fvmptd3 7011 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
153 | | iftrue 4526 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ 𝐽 → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) |
154 | 153 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → if(𝑘 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) |
155 | 152, 154 | eqtrd 2764 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘) = ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) |
156 | 155 | oveq2d 7417 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) |
157 | 156 | mpteq2dva 5238 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) |
158 | 148, 157 | eqtrd 2764 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) |
159 | 158 | oveq2d 7417 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽)) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) |
160 | 146, 159 | oveq12d 7419 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ (𝐼 ∖ 𝐽)))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg ((𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))) ↾ 𝐽))) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
161 | 54 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))):(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
162 | 4, 20, 30 | mplsca 21882 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) = (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
163 | 162 | fveq2d 6885 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
164 | 163 | fveq2d 6885 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 →
(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) =
(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
165 | 164 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) →
(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) =
(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
166 | 165 | oveqd 7418 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
167 | | eqid 2724 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(mulGrp‘((𝐼
∖ 𝐽) mPoly 𝑅)) = (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
168 | 167, 53 | mgpbas 20035 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘((𝐼
∖ 𝐽) mPoly 𝑅)) =
(Base‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
169 | | eqid 2724 |
. . . . . . . . . . . . . . . . . . 19
⊢
(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) =
(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
170 | 167 | crngmgp 20136 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd) |
171 | 22, 170 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd) |
172 | 171 | cmnmndd 19714 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd) |
173 | 172 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Mnd) |
174 | 135, 88 | sylan2 592 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑔‘𝑘) ∈
ℕ0) |
175 | 21 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝐼 ∖ 𝐽) ∈ V) |
176 | 57 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring) |
177 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ (𝐼 ∖ 𝐽)) |
178 | 3, 56, 53, 175, 176, 177 | mvrcl 21861 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
179 | 168, 169,
173, 174, 178 | mulgnn0cld 19012 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
180 | 166, 179 | eqeltrrd 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
181 | 161, 180 | cofmpt 7122 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
182 | | eqid 2724 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
183 | 182, 17 | rhmmhm 20371 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ ((Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) RingHom (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
184 | 28, 183 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
185 | 184 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
186 | 162 | fveq2d 6885 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
187 | 186 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
188 | 178, 187 | eleqtrd 2827 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
189 | | eqid 2724 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
190 | 182, 189 | mgpbas 20035 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) =
(Base‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
191 | | eqid 2724 |
. . . . . . . . . . . . . . . . . . 19
⊢
(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) =
(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
192 | 190, 191,
18 | mhmmulg 19032 |
. . . . . . . . . . . . . . . . . 18
⊢
(((algSc‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ (𝑔‘𝑘) ∈ ℕ0 ∧ (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
193 | 185, 174,
188, 192 | syl3anc 1368 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
194 | 193 | mpteq2dva 5238 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
195 | 181, 194 | eqtrd 2764 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
196 | 195 | oveq2d 7417 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg
((algSc‘(𝐽 mPoly
((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))) |
197 | | eqid 2724 |
. . . . . . . . . . . . . . 15
⊢
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) =
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
198 | 162, 22 | eqeltrrd 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing) |
199 | 182 | crngmgp 20136 |
. . . . . . . . . . . . . . . . 17
⊢
((Scalar‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ CRing →
(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd) |
200 | 198, 199 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd) |
201 | 200 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ CMnd) |
202 | 84 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd) |
203 | 21 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐼 ∖ 𝐽) ∈ V) |
204 | 184 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) MndHom (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
205 | 37, 48 | eqeltrrd 2826 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Ring) |
206 | 182 | ringmgp 20134 |
. . . . . . . . . . . . . . . . . . 19
⊢
((Scalar‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Ring →
(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ Mnd) |
207 | 205, 206 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ Mnd) |
208 | 207 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ Mnd) |
209 | 190, 191,
208, 174, 188 | mulgnn0cld 19012 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
210 | 209 | fmpttd 7106 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))):(𝐼 ∖ 𝐽)⟶(Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
211 | 21 | mptexd 7217 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ V) |
212 | 211 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ V) |
213 | | fvexd 6896 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) →
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ∈ V) |
214 | | funmpt 6576 |
. . . . . . . . . . . . . . . . 17
⊢ Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
215 | 214 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → Fun (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
216 | 119, 133,
123 | fmptssfisupp 9385 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) finSupp 0) |
217 | | ssidd 3997 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) supp 0) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) supp 0)) |
218 | 186 | eqimssd 4030 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ⊆ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
219 | 218 | sselda 3974 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
220 | 219 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → 𝑢 ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
221 | 190, 197,
191 | mulg0 18992 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 ∈
(Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) →
(0(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))𝑢) =
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
222 | 220, 221 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑢 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) →
(0(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))𝑢) =
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
223 | 101 | adantlr 712 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
224 | 217, 222,
174, 223, 123 | suppssov1 8177 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) supp
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) supp 0)) |
225 | 212, 213,
215, 216, 224 | fsuppsssuppgd 41557 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) finSupp
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
226 | 190, 197,
201, 202, 203, 204, 210, 225 | gsummhm 19848 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg
((algSc‘(𝐽 mPoly
((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))) |
227 | 196, 226 | eqtr3d 2766 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) = ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))) |
228 | 227 | oveq1d 7416 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
229 | 25 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg) |
230 | 58 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mVar 𝑅):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
231 | 230 | ffvelcdmda 7076 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
232 | 217, 222,
174, 231, 123 | suppssov1 8177 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) supp
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) supp 0)) |
233 | 212, 213,
215, 216, 232 | fsuppsssuppgd 41557 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) finSupp
(0g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
234 | 190, 197,
201, 203, 210, 233 | gsumcl 19825 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
235 | 20 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝐽 ∈ V) |
236 | 84 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ Mnd) |
237 | 150, 88 | syldan 590 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (𝑔‘𝑘) ∈
ℕ0) |
238 | 77, 18, 236, 237, 97 | mulgnn0cld 19012 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
239 | 238 | fmpttd 7106 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))):𝐽⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
240 | 20 | mptexd 7217 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) ∈ V) |
241 | 240 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) ∈ V) |
242 | | funmpt 6576 |
. . . . . . . . . . . . . . . 16
⊢ Fun
(𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) |
243 | 242 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → Fun (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) |
244 | 119, 147,
123 | fmptssfisupp 9385 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) finSupp 0) |
245 | | ssidd 3997 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) supp 0) ⊆ ((𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) supp 0)) |
246 | 245, 122,
237, 97, 123 | suppssov1 8177 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) supp
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ⊆ ((𝑘 ∈ 𝐽 ↦ (𝑔‘𝑘)) supp 0)) |
247 | 241, 113,
243, 244, 246 | fsuppsssuppgd 41557 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) finSupp
(0g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
248 | 77, 78, 82, 235, 239, 247 | gsumcl 19825 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
249 | | eqid 2724 |
. . . . . . . . . . . . . 14
⊢ (
·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = ( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
250 | 5, 26, 189, 16, 19, 249 | asclmul1 21748 |
. . . . . . . . . . . . 13
⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg ∧
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
251 | 229, 234,
248, 250 | syl3anc 1368 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
252 | 228, 251 | eqtrd 2764 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
253 | 132, 160,
252 | 3eqtrd 2768 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) = (((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
254 | 164 | oveqd 7418 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
255 | 254 | mpteq2dv 5240 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
256 | 163, 255 | oveq12d 7419 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
257 | 256 | eqcomd 2730 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
258 | 257 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
259 | 258 | oveq1d 7416 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) →
(((mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠 ‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
260 | 253, 259 | eqtrd 2764 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) |
261 | 76, 260 | oveq12d 7419 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))) = (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) |
262 | 73, 74 | ffvelcdmd 7077 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
263 | 186 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
264 | 262, 263 | eleqtrd 2827 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
265 | 4, 20, 48 | mpllmodd 41605 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod) |
266 | 265 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod) |
267 | | eqid 2724 |
. . . . . . . . . . . 12
⊢
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) =
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
268 | 171 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd) |
269 | 179 | fmpttd 7106 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))):(𝐼 ∖ 𝐽)⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
270 | 21 | mptexd 7217 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ V) |
271 | 270 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) ∈ V) |
272 | | fvexd 6896 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) →
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V) |
273 | | funmpt 6576 |
. . . . . . . . . . . . . 14
⊢ Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
274 | 273 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → Fun (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
275 | 168, 267,
169 | mulg0 18992 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) →
(0(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑒) =
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
276 | 275 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑒 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) →
(0(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑒) =
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
277 | 217, 276,
174, 178, 123 | suppssov1 8177 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) supp
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑔‘𝑘)) supp 0)) |
278 | 271, 272,
274, 216, 277 | fsuppsssuppgd 41557 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) finSupp
(0g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
279 | 168, 267,
268, 203, 269, 278 | gsumcl 19825 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
280 | 279, 263 | eleqtrd 2827 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
281 | 16, 26, 249, 189, 266, 280, 248 | lmodvscld 20715 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
282 | 5, 26, 189, 16, 19, 249 | asclmul1 21748 |
. . . . . . . . 9
⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ AssAlg ∧ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) ∈ (Base‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∧ (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) |
283 | 229, 264,
281, 282 | syl3anc 1368 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) |
284 | 261, 283 | eqtrd 2764 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) |
285 | 284 | mpteq2dva 5238 |
. . . . . 6
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘)))))) = (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) |
286 | 285 | oveq2d 7417 |
. . . . 5
⊢ (𝜑 → ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ (algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∘ 𝐹)‘𝑔)(.r‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐼 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝑧 ∈ 𝐼 ↦ if(𝑧 ∈ 𝐽, ((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑧), ((algSc‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑧))))‘𝑘))))))) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) |
287 | 11, 66, 286 | 3eqtrd 2768 |
. . . 4
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) |
288 | 287 | fveq1d 6883 |
. . 3
⊢ (𝜑 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽)) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))) |
289 | 288 | fveq1d 6883 |
. 2
⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
290 | | eqid 2724 |
. . . 4
⊢
(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
291 | 48 | ringcmnd 20173 |
. . . 4
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CMnd) |
292 | 8 | crnggrpd 20142 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Grp) |
293 | 292 | grpmndd 18866 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Mnd) |
294 | | ovex 7434 |
. . . . . 6
⊢
(ℕ0 ↑m 𝐼) ∈ V |
295 | 15, 294 | rabex2 5324 |
. . . . 5
⊢ 𝐷 ∈ V |
296 | 295 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ V) |
297 | | eqid 2724 |
. . . . . 6
⊢ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈
Fin} |
298 | | eqid 2724 |
. . . . . 6
⊢ (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
299 | | difssd 4124 |
. . . . . . 7
⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼) |
300 | | selvvvval.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝐷) |
301 | 15, 297, 7, 299, 300 | psrbagres 41604 |
. . . . . 6
⊢ (𝜑 → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈
Fin}) |
302 | 3, 53, 297, 298, 21, 292, 301 | mplmapghm 41617 |
. . . . 5
⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅)) |
303 | | ghmmhm 19141 |
. . . . 5
⊢ ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) GrpHom 𝑅) → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) MndHom 𝑅)) |
304 | 302, 303 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∈ (((𝐼 ∖ 𝐽) mPoly 𝑅) MndHom 𝑅)) |
305 | | eqid 2724 |
. . . . . . . 8
⊢ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin} |
306 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
307 | 4, 53, 16, 305, 306 | mplelf 21867 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → 𝑤:{𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}⟶(Base‘((𝐼
∖ 𝐽) mPoly 𝑅))) |
308 | 15, 305, 7, 9, 300 | psrbagres 41604 |
. . . . . . . 8
⊢ (𝜑 → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}) |
309 | 308 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}) |
310 | 307, 309 | ffvelcdmd 7077 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) → (𝑤‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
311 | 310 | fmpttd 7106 |
. . . . 5
⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))):(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
312 | 16, 26, 249, 189, 266, 264, 281 | lmodvscld 20715 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
313 | 312 | fmpttd 7106 |
. . . . 5
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))):𝐷⟶(Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
314 | 311, 313 | fcod 6733 |
. . . 4
⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))):𝐷⟶(Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
315 | | fvexd 6896 |
. . . . 5
⊢ (𝜑 →
(0g‘((𝐼
∖ 𝐽) mPoly 𝑅)) ∈ V) |
316 | 23 | crngringd 20141 |
. . . . . 6
⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring) |
317 | | eqid 2724 |
. . . . . . 7
⊢
(0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
318 | 16, 317 | ring0cl 20156 |
. . . . . 6
⊢ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ Ring →
(0g‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
319 | 316, 318 | syl 17 |
. . . . 5
⊢ (𝜑 →
(0g‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
320 | | ssidd 3997 |
. . . . 5
⊢ (𝜑 → (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ⊆ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
321 | 295 | mptex 7216 |
. . . . . . 7
⊢ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) ∈ V |
322 | 321 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) ∈ V) |
323 | | fvexd 6896 |
. . . . . 6
⊢ (𝜑 →
(0g‘(𝐽
mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V) |
324 | | funmpt 6576 |
. . . . . . 7
⊢ Fun
(𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) |
325 | 324 | a1i 11 |
. . . . . 6
⊢ (𝜑 → Fun (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) |
326 | 295 | mptex 7216 |
. . . . . . . 8
⊢ (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) ∈ V |
327 | 326 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) ∈ V) |
328 | | fvexd 6896 |
. . . . . . 7
⊢ (𝜑 →
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ∈ V) |
329 | | funmpt 6576 |
. . . . . . . 8
⊢ Fun
(𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) |
330 | 329 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → Fun (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)))) |
331 | | eqid 2724 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
332 | 1, 2, 331, 10, 8 | mplelsfi 21864 |
. . . . . . 7
⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅)) |
333 | | ssidd 3997 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 supp (0g‘𝑅)) ⊆ (𝐹 supp (0g‘𝑅))) |
334 | | fvexd 6896 |
. . . . . . . . . . 11
⊢ (𝜑 → (0g‘𝑅) ∈ V) |
335 | 68, 333, 10, 334 | suppssrg 8176 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑅)))) → (𝐹‘𝑔) = (0g‘𝑅)) |
336 | 335 | fveq2d 6885 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) = ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0g‘𝑅))) |
337 | 3, 31, 331, 290, 21, 57 | mplascl0 41615 |
. . . . . . . . . . 11
⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0g‘𝑅)) =
(0g‘((𝐼
∖ 𝐽) mPoly 𝑅))) |
338 | 162 | fveq2d 6885 |
. . . . . . . . . . 11
⊢ (𝜑 →
(0g‘((𝐼
∖ 𝐽) mPoly 𝑅)) =
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
339 | 337, 338 | eqtrd 2764 |
. . . . . . . . . 10
⊢ (𝜑 → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0g‘𝑅)) =
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
340 | 339 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(0g‘𝑅)) =
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
341 | 336, 340 | eqtrd 2764 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑅)))) → ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔)) =
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
342 | 341, 296 | suppss2 8180 |
. . . . . . 7
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) supp
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ⊆ (𝐹 supp (0g‘𝑅))) |
343 | 327, 328,
330, 332, 342 | fsuppsssuppgd 41557 |
. . . . . 6
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) finSupp
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
344 | | ssidd 3997 |
. . . . . . 7
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) supp
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))) ⊆ ((𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) supp
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
345 | | eqid 2724 |
. . . . . . . . 9
⊢
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) =
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
346 | 16, 26, 249, 345, 317 | lmod0vs 20731 |
. . . . . . . 8
⊢ (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ LMod ∧ 𝑓 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) →
((0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑓) = (0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
347 | 265, 346 | sylan 579 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) →
((0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))𝑓) = (0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
348 | 344, 347,
262, 281, 328 | suppssov1 8177 |
. . . . . 6
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) supp (0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) ⊆ ((𝑔 ∈ 𝐷 ↦ ((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))) supp
(0g‘(Scalar‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))))) |
349 | 322, 323,
325, 343, 348 | fsuppsssuppgd 41557 |
. . . . 5
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) finSupp (0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
350 | | eqid 2724 |
. . . . . . . 8
⊢ (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) = (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) |
351 | 22 | crnggrpd 20142 |
. . . . . . . 8
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Grp) |
352 | 4, 16, 305, 350, 20, 351, 308 | mplmapghm 41617 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) GrpHom ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
353 | | ghmmhm 19141 |
. . . . . . 7
⊢ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) GrpHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
354 | 352, 353 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅))) |
355 | 317, 290 | mhm0 18714 |
. . . . . 6
⊢ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∈ ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) MndHom ((𝐼 ∖ 𝐽) mPoly 𝑅)) → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘(0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
356 | 354, 355 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘(0g‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
357 | 315, 319,
313, 311, 320, 296, 46, 349, 356 | fsuppcor 9395 |
. . . 4
⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) finSupp (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
358 | 53, 290, 291, 293, 296, 304, 314, 357 | gsummhm 19848 |
. . 3
⊢ (𝜑 → (𝑅 Σg ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))‘(((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))))) |
359 | | fveq1 6880 |
. . . 4
⊢ (𝑣 = (((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) → (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
360 | 53, 290, 291, 296, 314, 357 | gsumcl 19825 |
. . . 4
⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
361 | | fvexd 6896 |
. . . 4
⊢ (𝜑 → ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) ∈ V) |
362 | 298, 359,
360, 361 | fvmptd3 7011 |
. . 3
⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))‘(((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
363 | 316 | ringcmnd 20173 |
. . . . . 6
⊢ (𝜑 → (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ CMnd) |
364 | 351 | grpmndd 18866 |
. . . . . 6
⊢ (𝜑 → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Mnd) |
365 | 16, 317, 363, 364, 296, 354, 313, 349 | gsummhm 19848 |
. . . . 5
⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) |
366 | | fveq1 6880 |
. . . . . 6
⊢ (𝑤 = ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) → (𝑤‘(𝑌 ↾ 𝐽)) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))) |
367 | 16, 317, 363, 296, 313, 349 | gsumcl 19825 |
. . . . . 6
⊢ (𝜑 → ((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
368 | | fvexd 6896 |
. . . . . 6
⊢ (𝜑 → (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽)) ∈ V) |
369 | 350, 366,
367, 368 | fvmptd3 7011 |
. . . . 5
⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))‘((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))) |
370 | 365, 369 | eqtrd 2764 |
. . . 4
⊢ (𝜑 → (((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))) |
371 | 370 | fveq1d 6883 |
. . 3
⊢ (𝜑 → ((((𝐼 ∖ 𝐽) mPoly 𝑅) Σg ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
372 | 358, 362,
371 | 3eqtrrd 2769 |
. 2
⊢ (𝜑 → ((((𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝑅 Σg ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))))) |
373 | 4, 53, 16, 305, 312 | mplelf 21867 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))):{𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}⟶(Base‘((𝐼
∖ 𝐽) mPoly 𝑅))) |
374 | 308 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}) |
375 | 373, 374 | ffvelcdmd 7077 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
376 | | eqidd 2725 |
. . . . . . 7
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))) = (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) |
377 | | eqidd 2725 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) = (𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽)))) |
378 | | fveq1 6880 |
. . . . . . 7
⊢ (𝑤 = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))) → (𝑤‘(𝑌 ↾ 𝐽)) = ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))) |
379 | 312, 376,
377, 378 | fmptco 7119 |
. . . . . 6
⊢ (𝜑 → ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))) = (𝑔 ∈ 𝐷 ↦ ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)))) |
380 | | eqidd 2725 |
. . . . . 6
⊢ (𝜑 → (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))))) |
381 | | fveq1 6880 |
. . . . . 6
⊢ (𝑣 = ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) → (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
382 | 375, 379,
380, 381 | fmptco 7119 |
. . . . 5
⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (𝑔 ∈ 𝐷 ↦ (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))) |
383 | | eqid 2724 |
. . . . . . . . . 10
⊢
(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (.r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
384 | 4, 249, 53, 16, 383, 305, 262, 281, 374 | mplvscaval 21885 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽)))) |
385 | 4, 249, 53, 16, 383, 305, 279, 248, 374 | mplvscaval 21885 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽)) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) |
386 | 385 | oveq2d 7417 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))‘(𝑌 ↾ 𝐽))) = (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))) |
387 | 30 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg) |
388 | 35 | fveq2d 6885 |
. . . . . . . . . . . 12
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
389 | 388 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (Base‘𝑅) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
390 | 74, 389 | eleqtrd 2827 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐹‘𝑔) ∈ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
391 | 48 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ Ring) |
392 | 4, 53, 16, 305, 248 | mplelf 21867 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))):{𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}⟶(Base‘((𝐼
∖ 𝐽) mPoly 𝑅))) |
393 | 392, 374 | ffvelcdmd 7077 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
394 | 53, 383, 391, 279, 393 | ringcld 20152 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
395 | | eqid 2724 |
. . . . . . . . . . 11
⊢
(Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
396 | | eqid 2724 |
. . . . . . . . . . 11
⊢ (
·𝑠 ‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
397 | 31, 32, 395, 53, 383, 396 | asclmul1 21748 |
. . . . . . . . . 10
⊢ ((((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ AssAlg ∧ (𝐹‘𝑔) ∈ (Base‘(Scalar‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∧ (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) = ((𝐹‘𝑔)( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))) |
398 | 387, 390,
394, 397 | syl3anc 1368 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) = ((𝐹‘𝑔)( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))) |
399 | 384, 386,
398 | 3eqtrd 2768 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽)) = ((𝐹‘𝑔)( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))) |
400 | 399 | fveq1d 6883 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (((𝐹‘𝑔)( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
401 | | eqid 2724 |
. . . . . . . 8
⊢
(.r‘𝑅) = (.r‘𝑅) |
402 | 301 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈
Fin}) |
403 | 3, 396, 67, 53, 401, 297, 74, 394, 402 | mplvscaval 21885 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝐹‘𝑔)( ·𝑠
‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((𝐹‘𝑔)(.r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))) |
404 | | eqid 2724 |
. . . . . . . . . . . . . 14
⊢
(1r‘𝑅) = (1r‘𝑅) |
405 | 8 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ CRing) |
406 | 15, 297, 83, 133, 70 | psrbagres 41604 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈
Fin}) |
407 | 3, 297, 331, 404, 203, 167, 169, 56, 405, 406 | mplcoe2 21906 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
408 | 177 | fvresd 6901 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑔‘𝑘)) |
409 | 408 | oveq1d 7416 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) |
410 | 409 | mpteq2dva 5238 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) |
411 | 410 | oveq2d 7417 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑔 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘)))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
412 | 407, 411 | eqtrd 2764 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅))) = ((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))) |
413 | | eqid 2724 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
414 | | eqeq1 2728 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑌 ↾ 𝐽) → (𝑗 = (𝑔 ↾ 𝐽) ↔ (𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽))) |
415 | 414 | ifbid 4543 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑌 ↾ 𝐽) → if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
416 | | fvexd 6896 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V) |
417 | | fvexd 6896 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ∈ V) |
418 | 416, 417 | ifcld 4566 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) ∈ V) |
419 | 413, 415,
374, 418 | fvmptd3 7011 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ 𝐽)) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
420 | | eqid 2724 |
. . . . . . . . . . . . . . . 16
⊢
(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)) |
421 | 22 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ CRing) |
422 | 15, 305, 83, 147, 70 | psrbagres 41604 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 ↾ 𝐽) ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈
Fin}) |
423 | 4, 305, 290, 420, 235, 17, 18, 47, 421, 422 | mplcoe2 21906 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) |
424 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → 𝑘 ∈ 𝐽) |
425 | 424 | fvresd 6901 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → ((𝑔 ↾ 𝐽)‘𝑘) = (𝑔‘𝑘)) |
426 | 425 | oveq1d 7416 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐷) ∧ 𝑘 ∈ 𝐽) → (((𝑔 ↾ 𝐽)‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)) = ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) |
427 | 426 | mpteq2dva 5238 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))) = (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) |
428 | 427 | oveq2d 7417 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ (((𝑔 ↾ 𝐽)‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) |
429 | 423, 428 | eqtrd 2764 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = ((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))) |
430 | 429 | fveq1d 6883 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑥 “ ℕ) ∈ Fin} ↦
if(𝑗 = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ 𝐽)) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) |
431 | 419, 430 | eqtr3d 2766 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) |
432 | 412, 431 | oveq12d 7419 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))) |
433 | 432 | eqcomd 2730 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽))) = ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))) |
434 | 433 | fveq1d 6883 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
435 | 434 | oveq2d 7417 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = ((𝐹‘𝑔)(.r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))))) |
436 | | ovif2 7499 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))) |
437 | 436 | fveq1i 6882 |
. . . . . . . . . . 11
⊢ (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) |
438 | | iffv 6898 |
. . . . . . . . . . . 12
⊢
(if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))), (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
439 | | eqeq1 2728 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → (𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)) ↔ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)))) |
440 | 439 | ifbid 4543 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (𝑌 ↾ (𝐼 ∖ 𝐽)) → if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)) = if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅))) |
441 | 57 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑅 ∈ Ring) |
442 | 3, 53, 331, 404, 297, 203, 441, 406 | mplmon 21900 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅))) ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
443 | 53, 383, 420, 391, 442 | ringridmd 20162 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))) |
444 | | fvexd 6896 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (1r‘𝑅) ∈ V) |
445 | | fvexd 6896 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0g‘𝑅) ∈ V) |
446 | 444, 445 | ifcld 4566 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)) ∈ V) |
447 | 440, 443,
402, 446 | fvmptd4 41546 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅))) |
448 | 53, 383, 290, 391, 442 | ringrzd 20185 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) = (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))) |
449 | 448 | fveq1d 6883 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = ((0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
450 | 3, 297, 331, 290, 21, 292 | mpl0 21875 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(0g‘((𝐼
∖ 𝐽) mPoly 𝑅)) = ({𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
451 | 450 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)) = ({𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
452 | 451 | fveq1d 6883 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (({𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) |
453 | | fvex 6894 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑅) ∈ V |
454 | 453 | fvconst2 7197 |
. . . . . . . . . . . . . . 15
⊢ ((𝑌 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} → (({𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0g‘𝑅)) |
455 | 402, 454 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (({𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0g‘𝑅)) |
456 | 449, 452,
455 | 3eqtrd 2768 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (0g‘𝑅)) |
457 | 447, 456 | ifeq12d 4541 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽))), (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) |
458 | 438, 457 | eqtrid 2776 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(1r‘((𝐼 ∖ 𝐽) mPoly 𝑅))), ((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) |
459 | 437, 458 | eqtrid 2776 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) |
460 | 459 | oveq2d 7417 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = ((𝐹‘𝑔)(.r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅)))) |
461 | | ifan 4573 |
. . . . . . . . . . 11
⊢
if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1r‘𝑅), (0g‘𝑅)) = if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅)) |
462 | 461 | oveq2i 7412 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑔)(.r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1r‘𝑅), (0g‘𝑅))) = ((𝐹‘𝑔)(.r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) |
463 | 15 | psrbagf 21780 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ0) |
464 | 300, 463 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
465 | 464 | ffnd 6708 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑌 Fn 𝐼) |
466 | 465 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn 𝐼) |
467 | | undif 4473 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
468 | 9, 467 | sylib 217 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
469 | 468 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
470 | 469 | fneq2d 6633 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑌 Fn 𝐼)) |
471 | 466, 470 | mpbird 257 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽))) |
472 | 87 | ffnd 6708 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn 𝐼) |
473 | 469 | fneq2d 6633 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑔 Fn 𝐼)) |
474 | 472, 473 | mpbird 257 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → 𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽))) |
475 | | eqfnun 7028 |
. . . . . . . . . . . . . 14
⊢ ((𝑌 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ∧ 𝑔 Fn (𝐽 ∪ (𝐼 ∖ 𝐽))) → (𝑌 = 𝑔 ↔ ((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))) |
476 | 471, 474,
475 | syl2anc 583 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (𝑌 = 𝑔 ↔ ((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))))) |
477 | 476 | ifbid 4543 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (1r‘𝑅), (0g‘𝑅)) = if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1r‘𝑅), (0g‘𝑅))) |
478 | 477 | oveq2d 7417 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)if(𝑌 = 𝑔, (1r‘𝑅), (0g‘𝑅))) = ((𝐹‘𝑔)(.r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1r‘𝑅), (0g‘𝑅)))) |
479 | | ovif2 7499 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑔)(.r‘𝑅)if(𝑌 = 𝑔, (1r‘𝑅), (0g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)), ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅))) |
480 | 478, 479 | eqtr3di 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)if(((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽) ∧ (𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽))), (1r‘𝑅), (0g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)), ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅)))) |
481 | 462, 480 | eqtr3id 2778 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), if((𝑌 ↾ (𝐼 ∖ 𝐽)) = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)), ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅)))) |
482 | 460, 481 | eqtrd 2764 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)(((𝑖 ∈ {𝑦 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑦 “ ℕ) ∈ Fin} ↦
if(𝑖 = (𝑔 ↾ (𝐼 ∖ 𝐽)), (1r‘𝑅), (0g‘𝑅)))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))if((𝑌 ↾ 𝐽) = (𝑔 ↾ 𝐽), (1r‘((𝐼 ∖ 𝐽) mPoly 𝑅)), (0g‘((𝐼 ∖ 𝐽) mPoly 𝑅))))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if(𝑌 = 𝑔, ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)), ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅)))) |
483 | 67, 401, 404, 441, 74 | ringridmd 20162 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)) = (𝐹‘𝑔)) |
484 | 67, 401, 331, 441, 74 | ringrzd 20185 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
485 | 483, 484 | ifeq12d 4541 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, ((𝐹‘𝑔)(.r‘𝑅)(1r‘𝑅)), ((𝐹‘𝑔)(.r‘𝑅)(0g‘𝑅))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) |
486 | 435, 482,
485 | 3eqtrd 2768 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → ((𝐹‘𝑔)(.r‘𝑅)((((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))(.r‘((𝐼 ∖ 𝐽) mPoly 𝑅))(((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))‘(𝑌 ↾ 𝐽)))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) |
487 | 400, 403,
486 | 3eqtrd 2768 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) |
488 | 487 | mpteq2dva 5238 |
. . . . 5
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ (((((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) = (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) |
489 | 382, 488 | eqtrd 2764 |
. . . 4
⊢ (𝜑 → ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘))))))))) = (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) |
490 | 489 | oveq2d 7417 |
. . 3
⊢ (𝜑 → (𝑅 Σg ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = (𝑅 Σg (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))))) |
491 | 57 | ringcmnd 20173 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ CMnd) |
492 | 67, 331 | ring0cl 20156 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring →
(0g‘𝑅)
∈ (Base‘𝑅)) |
493 | 57, 492 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) |
494 | 493 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → (0g‘𝑅) ∈ (Base‘𝑅)) |
495 | 74, 494 | ifcld 4566 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐷) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) ∈ (Base‘𝑅)) |
496 | 495 | fmpttd 7106 |
. . . 4
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))):𝐷⟶(Base‘𝑅)) |
497 | | eldifsnneq 4786 |
. . . . . . . 8
⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑔 = 𝑌) |
498 | 497 | neqcomd 2734 |
. . . . . . 7
⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑌 = 𝑔) |
499 | 498 | iffalsed 4531 |
. . . . . 6
⊢ (𝑔 ∈ (𝐷 ∖ {𝑌}) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) = (0g‘𝑅)) |
500 | 499 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 ∖ {𝑌})) → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) = (0g‘𝑅)) |
501 | 500, 296 | suppss2 8180 |
. . . 4
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) supp (0g‘𝑅)) ⊆ {𝑌}) |
502 | 296 | mptexd 7217 |
. . . . 5
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) ∈ V) |
503 | | funmpt 6576 |
. . . . . 6
⊢ Fun
(𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) |
504 | 503 | a1i 11 |
. . . . 5
⊢ (𝜑 → Fun (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) |
505 | | snfi 9040 |
. . . . . . 7
⊢ {𝑌} ∈ Fin |
506 | 505 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝑌} ∈ Fin) |
507 | 506, 501 | ssfid 9263 |
. . . . 5
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) supp (0g‘𝑅)) ∈ Fin) |
508 | 502, 493,
504, 507 | isfsuppd 9362 |
. . . 4
⊢ (𝜑 → (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) finSupp (0g‘𝑅)) |
509 | 67, 331, 491, 296, 496, 501, 508 | gsumres 19823 |
. . 3
⊢ (𝜑 → (𝑅 Σg ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) ↾ {𝑌})) = (𝑅 Σg (𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))))) |
510 | 300 | snssd 4804 |
. . . . . 6
⊢ (𝜑 → {𝑌} ⊆ 𝐷) |
511 | 510 | resmptd 6030 |
. . . . 5
⊢ (𝜑 → ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) ↾ {𝑌}) = (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) |
512 | 511 | oveq2d 7417 |
. . . 4
⊢ (𝜑 → (𝑅 Σg ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) ↾ {𝑌})) = (𝑅 Σg (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))))) |
513 | 68, 300 | ffvelcdmd 7077 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝑅)) |
514 | | iftrue 4526 |
. . . . . . . 8
⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) = (𝐹‘𝑔)) |
515 | | fveq2 6881 |
. . . . . . . . 9
⊢ (𝑔 = 𝑌 → (𝐹‘𝑔) = (𝐹‘𝑌)) |
516 | 515 | eqcoms 2732 |
. . . . . . . 8
⊢ (𝑌 = 𝑔 → (𝐹‘𝑔) = (𝐹‘𝑌)) |
517 | 514, 516 | eqtrd 2764 |
. . . . . . 7
⊢ (𝑌 = 𝑔 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) = (𝐹‘𝑌)) |
518 | 517 | eqcoms 2732 |
. . . . . 6
⊢ (𝑔 = 𝑌 → if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)) = (𝐹‘𝑌)) |
519 | 67, 518 | gsumsn 19864 |
. . . . 5
⊢ ((𝑅 ∈ Mnd ∧ 𝑌 ∈ 𝐷 ∧ (𝐹‘𝑌) ∈ (Base‘𝑅)) → (𝑅 Σg (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) = (𝐹‘𝑌)) |
520 | 293, 300,
513, 519 | syl3anc 1368 |
. . . 4
⊢ (𝜑 → (𝑅 Σg (𝑔 ∈ {𝑌} ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅)))) = (𝐹‘𝑌)) |
521 | 512, 520 | eqtrd 2764 |
. . 3
⊢ (𝜑 → (𝑅 Σg ((𝑔 ∈ 𝐷 ↦ if(𝑌 = 𝑔, (𝐹‘𝑔), (0g‘𝑅))) ↾ {𝑌})) = (𝐹‘𝑌)) |
522 | 490, 509,
521 | 3eqtr2d 2770 |
. 2
⊢ (𝜑 → (𝑅 Σg ((𝑣 ∈ (Base‘((𝐼 ∖ 𝐽) mPoly 𝑅)) ↦ (𝑣‘(𝑌 ↾ (𝐼 ∖ 𝐽)))) ∘ ((𝑤 ∈ (Base‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) ↦ (𝑤‘(𝑌 ↾ 𝐽))) ∘ (𝑔 ∈ 𝐷 ↦ (((algSc‘((𝐼 ∖ 𝐽) mPoly 𝑅))‘(𝐹‘𝑔))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))(((mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘((𝐼 ∖ 𝐽) mPoly 𝑅)))(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑘))))( ·𝑠
‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))((mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))) Σg (𝑘 ∈ 𝐽 ↦ ((𝑔‘𝑘)(.g‘(mulGrp‘(𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅))))((𝐽 mVar ((𝐼 ∖ 𝐽) mPoly 𝑅))‘𝑘)))))))))) = (𝐹‘𝑌)) |
523 | 289, 372,
522 | 3eqtrd 2768 |
1
⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌)) |