Step | Hyp | Ref
| Expression |
1 | | mpfind.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑄) |
2 | | mpfind.cq |
. . . . 5
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) |
3 | 1, 2 | eleqtrdi 2835 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
4 | 2 | mpfrcl 22092 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆))) |
5 | 1, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆))) |
6 | | eqid 2725 |
. . . . . . . 8
⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅) |
7 | | eqid 2725 |
. . . . . . . 8
⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅)) |
8 | | eqid 2725 |
. . . . . . . 8
⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅) |
9 | | eqid 2725 |
. . . . . . . 8
⊢ (𝑆 ↑s
(𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) |
10 | | mpfind.cb |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝑆) |
11 | 6, 7, 8, 9, 10 | evlsrhm 22095 |
. . . . . . 7
⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
12 | | eqid 2725 |
. . . . . . . 8
⊢
(Base‘(𝐼 mPoly
(𝑆 ↾s
𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) |
13 | | eqid 2725 |
. . . . . . . 8
⊢
(Base‘(𝑆
↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) |
14 | 12, 13 | rhmf 20462 |
. . . . . . 7
⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
15 | 5, 11, 14 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
16 | 15 | ffnd 6728 |
. . . . 5
⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
17 | | fvelrnb 6962 |
. . . . 5
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)) |
18 | 16, 17 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)) |
19 | 3, 18 | mpbid 231 |
. . 3
⊢ (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴) |
20 | 15 | ffund 6731 |
. . . . . 6
⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅)) |
21 | | eqid 2725 |
. . . . . . 7
⊢
(Base‘(𝑆
↾s 𝑅)) =
(Base‘(𝑆
↾s 𝑅)) |
22 | | eqid 2725 |
. . . . . . 7
⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅)) |
23 | | eqid 2725 |
. . . . . . 7
⊢
(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) |
24 | | eqid 2725 |
. . . . . . 7
⊢
(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) |
25 | | eqid 2725 |
. . . . . . 7
⊢
(algSc‘(𝐼
mPoly (𝑆
↾s 𝑅))) =
(algSc‘(𝐼 mPoly
(𝑆 ↾s
𝑅))) |
26 | 5 | simp1d 1139 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ∈ V) |
27 | 5 | simp2d 1140 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ∈ CRing) |
28 | 5 | simp3d 1141 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
29 | 8 | subrgcrng 20554 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝑅) ∈ CRing) |
30 | 27, 28, 29 | syl2anc 582 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ CRing) |
31 | | crngring 20223 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ↾s 𝑅) ∈ CRing → (𝑆 ↾s 𝑅) ∈ Ring) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring) |
33 | 7, 26, 32 | mplringd 22024 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
34 | 33 | adantr 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
35 | | simprl 769 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
36 | | elpreima 7070 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))) |
37 | 16, 36 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))) |
38 | 37 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))) |
39 | 35, 38 | mpbid 231 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})) |
40 | 39 | simpld 493 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
41 | | simprr 771 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
42 | | elpreima 7070 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) |
43 | 16, 42 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) |
44 | 43 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) |
45 | 41, 44 | mpbid 231 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})) |
46 | 45 | simpld 493 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
47 | 12, 23 | ringacl 20252 |
. . . . . . . . . 10
⊢ (((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
48 | 34, 40, 46, 47 | syl3anc 1368 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
49 | | rhmghm 20461 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
50 | 5, 11, 49 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
51 | 50 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
52 | | eqid 2725 |
. . . . . . . . . . . . 13
⊢
(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼))) =
(+g‘(𝑆
↑s (𝐵 ↑m 𝐼))) |
53 | 12, 23, 52 | ghmlin 19210 |
. . . . . . . . . . . 12
⊢ ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
54 | 51, 40, 46, 53 | syl3anc 1368 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
55 | 27 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑆 ∈ CRing) |
56 | | ovexd 7458 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐵 ↑m 𝐼) ∈ V) |
57 | 15 | adantr 479 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
58 | 57, 40 | ffvelcdmd 7098 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
59 | 57, 46 | ffvelcdmd 7098 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
60 | | mpfind.cp |
. . . . . . . . . . . 12
⊢ + =
(+g‘𝑆) |
61 | 9, 13, 55, 56, 58, 59, 60, 52 | pwsplusgval 17500 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
62 | 54, 61 | eqtrd 2765 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
63 | | simpl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝜑) |
64 | | fnfvelrn 7093 |
. . . . . . . . . . . . . 14
⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
65 | 16, 40, 64 | syl2an2r 683 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
66 | 65, 2 | eleqtrrdi 2836 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄) |
67 | | fvimacnvi 7064 |
. . . . . . . . . . . . 13
⊢ ((Fun
((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) |
68 | 20, 35, 67 | syl2an2r 683 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) |
69 | 66, 68 | jca 510 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})) |
70 | | fnfvelrn 7093 |
. . . . . . . . . . . . . 14
⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
71 | 16, 46, 70 | syl2an2r 683 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
72 | 71, 2 | eleqtrrdi 2836 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄) |
73 | | fvimacnvi 7064 |
. . . . . . . . . . . . 13
⊢ ((Fun
((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}) |
74 | 20, 41, 73 | syl2an2r 683 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}) |
75 | 72, 74 | jca 510 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})) |
76 | | fvex 6913 |
. . . . . . . . . . . 12
⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V |
77 | | fvex 6913 |
. . . . . . . . . . . 12
⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V |
78 | | eleq1 2813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)) |
79 | | vex 3465 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑓 ∈ V |
80 | | mpfind.wc |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏)) |
81 | 79, 80 | elab 3665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏) |
82 | | eleq1 2813 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})) |
83 | 81, 82 | bitr3id 284 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})) |
84 | 78, 83 | anbi12d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓 ∈ 𝑄 ∧ 𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))) |
85 | | eleq1 2813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)) |
86 | | vex 3465 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑔 ∈ V |
87 | | mpfind.wd |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂)) |
88 | 86, 87 | elab 3665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂) |
89 | | eleq1 2813 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})) |
90 | 88, 89 | bitr3id 284 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})) |
91 | 85, 90 | anbi12d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔 ∈ 𝑄 ∧ 𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) |
92 | 84, 91 | bi2anan9 636 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) ↔ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))) |
93 | 92 | anbi2d 628 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) ↔ (𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))))) |
94 | | ovex 7456 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∘f + 𝑔) ∈ V |
95 | | mpfind.we |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁)) |
96 | 94, 95 | elab 3665 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁) |
97 | | oveq12 7432 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓 ∘f + 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
98 | 97 | eleq1d 2810 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})) |
99 | 96, 98 | bitr3id 284 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝜁 ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})) |
100 | 93, 99 | imbi12d 343 |
. . . . . . . . . . . 12
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
101 | | mpfind.ad |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁) |
102 | 76, 77, 100, 101 | vtocl2 3546 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}) |
103 | 63, 69, 75, 102 | syl12anc 835 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}) |
104 | 62, 103 | eqeltrd 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}) |
105 | | elpreima 7070 |
. . . . . . . . . . 11
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
106 | 16, 105 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
107 | 106 | adantr 479 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
108 | 48, 104, 107 | mpbir2and 711 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
109 | 108 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
110 | 12, 24 | ringcl 20228 |
. . . . . . . . . 10
⊢ (((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
111 | 34, 40, 46, 110 | syl3anc 1368 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
112 | | eqid 2725 |
. . . . . . . . . . . . . . 15
⊢
(mulGrp‘(𝐼
mPoly (𝑆
↾s 𝑅))) =
(mulGrp‘(𝐼 mPoly
(𝑆 ↾s
𝑅))) |
113 | | eqid 2725 |
. . . . . . . . . . . . . . 15
⊢
(mulGrp‘(𝑆
↑s (𝐵 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))) |
114 | 112, 113 | rhmmhm 20456 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))) |
115 | 5, 11, 114 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))) |
116 | 115 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))) |
117 | 112, 12 | mgpbas 20118 |
. . . . . . . . . . . . 13
⊢
(Base‘(𝐼 mPoly
(𝑆 ↾s
𝑅))) =
(Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
118 | 112, 24 | mgpplusg 20116 |
. . . . . . . . . . . . 13
⊢
(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) =
(+g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
119 | | eqid 2725 |
. . . . . . . . . . . . . 14
⊢
(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) =
(.r‘(𝑆
↑s (𝐵 ↑m 𝐼))) |
120 | 113, 119 | mgpplusg 20116 |
. . . . . . . . . . . . 13
⊢
(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) =
(+g‘(mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
121 | 117, 118,
120 | mhmlin 18778 |
. . . . . . . . . . . 12
⊢ ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
122 | 116, 40, 46, 121 | syl3anc 1368 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
123 | | mpfind.ct |
. . . . . . . . . . . 12
⊢ · =
(.r‘𝑆) |
124 | 9, 13, 55, 56, 58, 59, 123, 119 | pwsmulrval 17501 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
125 | 122, 124 | eqtrd 2765 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
126 | | ovex 7456 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∘f · 𝑔) ∈ V |
127 | | mpfind.wf |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎)) |
128 | 126, 127 | elab 3665 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎) |
129 | | oveq12 7432 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓 ∘f · 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗))) |
130 | 129 | eleq1d 2810 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})) |
131 | 128, 130 | bitr3id 284 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝜎 ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})) |
132 | 93, 131 | imbi12d 343 |
. . . . . . . . . . . 12
⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
133 | | mpfind.mu |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎) |
134 | 76, 77, 132, 133 | vtocl2 3546 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}) |
135 | 63, 69, 75, 134 | syl12anc 835 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}) |
136 | 125, 135 | eqeltrd 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}) |
137 | | elpreima 7070 |
. . . . . . . . . . 11
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
138 | 16, 137 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
139 | 138 | adantr 479 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓}))) |
140 | 111, 136,
139 | mpbir2and 711 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
141 | 140 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
142 | 7 | mplassa 22023 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg) |
143 | 26, 30, 142 | syl2anc 582 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg) |
144 | | eqid 2725 |
. . . . . . . . . . . . 13
⊢
(Scalar‘(𝐼
mPoly (𝑆
↾s 𝑅))) =
(Scalar‘(𝐼 mPoly
(𝑆 ↾s
𝑅))) |
145 | 25, 144 | asclrhm 21879 |
. . . . . . . . . . . 12
⊢ ((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) RingHom (𝐼 mPoly (𝑆 ↾s 𝑅)))) |
146 | | eqid 2725 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
147 | 146, 12 | rhmf 20462 |
. . . . . . . . . . . 12
⊢
((algSc‘(𝐼
mPoly (𝑆
↾s 𝑅)))
∈ ((Scalar‘(𝐼
mPoly (𝑆
↾s 𝑅)))
RingHom (𝐼 mPoly (𝑆 ↾s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
148 | 143, 145,
147 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
149 | 148 | adantr 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
150 | 7, 26, 30 | mplsca 22014 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
151 | 150 | fveq2d 6904 |
. . . . . . . . . . . 12
⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) =
(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
152 | 151 | eleq2d 2811 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑖 ∈ (Base‘(𝑆 ↾s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))) |
153 | 152 | biimpa 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
154 | 149, 153 | ffvelcdmd 7098 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
155 | 26 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝐼 ∈ V) |
156 | 27 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑆 ∈ CRing) |
157 | 28 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆)) |
158 | 10 | subrgss 20551 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
159 | 8, 10 | ressbas2 17246 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅))) |
160 | 28, 158, 159 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅))) |
161 | 160 | eleq2d 2811 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑖 ∈ 𝑅 ↔ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅)))) |
162 | 161 | biimpar 476 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ 𝑅) |
163 | 6, 7, 8, 10, 25, 155, 156, 157, 162 | evlssca 22096 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) = ((𝐵 ↑m 𝐼) × {𝑖})) |
164 | | mpfind.co |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒) |
165 | 164 | ralrimiva 3135 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒) |
166 | | ovex 7456 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐵 ↑m 𝐼) ∈ V |
167 | | vsnex 5434 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓} ∈ V |
168 | 166, 167 | xpex 7760 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐵 ↑m 𝐼) × {𝑓}) ∈ V |
169 | | mpfind.wa |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒)) |
170 | 168, 169 | elab 3665 |
. . . . . . . . . . . . . . 15
⊢ (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒) |
171 | | sneq 4642 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖}) |
172 | 171 | xpeq2d 5711 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = 𝑖 → ((𝐵 ↑m 𝐼) × {𝑓}) = ((𝐵 ↑m 𝐼) × {𝑖})) |
173 | 172 | eleq1d 2810 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑖 → (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})) |
174 | 170, 173 | bitr3id 284 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})) |
175 | 174 | cbvralvw 3224 |
. . . . . . . . . . . . 13
⊢
(∀𝑓 ∈
𝑅 𝜒 ↔ ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}) |
176 | 165, 175 | sylib 217 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}) |
177 | 176 | r19.21bi 3238 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}) |
178 | 162, 177 | syldan 589 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}) |
179 | 163, 178 | eqeltrd 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓}) |
180 | | elpreima 7070 |
. . . . . . . . . . 11
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
181 | 16, 180 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
182 | 181 | adantr 479 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
183 | 154, 179,
182 | mpbir2and 711 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
184 | 183 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
185 | 26 | adantr 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ V) |
186 | 32 | adantr 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 ↾s 𝑅) ∈ Ring) |
187 | | simpr 483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼) |
188 | 7, 22, 12, 185, 186, 187 | mvrcl 21993 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
189 | 27 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing) |
190 | 28 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ (SubRing‘𝑆)) |
191 | 6, 22, 8, 10, 185, 189, 190, 187 | evlsvar 22097 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖))) |
192 | | mpfind.pr |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃) |
193 | 166 | mptex 7239 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ V |
194 | | mpfind.wb |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃)) |
195 | 193, 194 | elab 3665 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃) |
196 | 192, 195 | sylibr 233 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓}) |
197 | 196 | ralrimiva 3135 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓}) |
198 | | fveq2 6900 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖)) |
199 | 198 | mpteq2dv 5254 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖))) |
200 | 199 | eleq1d 2810 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})) |
201 | 200 | cbvralvw 3224 |
. . . . . . . . . . . 12
⊢
(∀𝑓 ∈
𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}) |
202 | 197, 201 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}) |
203 | 202 | r19.21bi 3238 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}) |
204 | 191, 203 | eqeltrd 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓}) |
205 | | elpreima 7070 |
. . . . . . . . . . 11
⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
206 | 16, 205 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
207 | 206 | adantr 479 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓}))) |
208 | 188, 204,
207 | mpbir2and 711 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
209 | 208 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
210 | | simpr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
211 | 26 | adantr 479 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝐼 ∈ V) |
212 | 30 | adantr 479 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑆 ↾s 𝑅) ∈ CRing) |
213 | 21, 22, 7, 23, 24, 25, 12, 109, 141, 184, 209, 210, 211, 212 | mplind 22075 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) |
214 | | fvimacnvi 7064 |
. . . . . 6
⊢ ((Fun
((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓}) |
215 | 20, 213, 214 | syl2an2r 683 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓}) |
216 | | eleq1 2813 |
. . . . 5
⊢ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})) |
217 | 215, 216 | syl5ibcom 244 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓})) |
218 | 217 | rexlimdva 3144 |
. . 3
⊢ (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓})) |
219 | 19, 218 | mpd 15 |
. 2
⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) |
220 | | mpfind.wg |
. . . 4
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌)) |
221 | 220 | elabg 3663 |
. . 3
⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌)) |
222 | 1, 221 | syl 17 |
. 2
⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌)) |
223 | 219, 222 | mpbid 231 |
1
⊢ (𝜑 → 𝜌) |