Proof of Theorem cpmadugsumlemC
Step | Hyp | Ref
| Expression |
1 | | eqid 2740 |
. . 3
⊢
(Base‘𝑌) =
(Base‘𝑌) |
2 | | eqid 2740 |
. . 3
⊢
(0g‘𝑌) = (0g‘𝑌) |
3 | | cpmadugsum.r |
. . 3
⊢ × =
(.r‘𝑌) |
4 | | crngring 20272 |
. . . . . . . 8
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
5 | | cpmadugsum.p |
. . . . . . . . 9
⊢ 𝑃 = (Poly1‘𝑅) |
6 | 5 | ply1ring 22270 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
7 | 4, 6 | syl 17 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
8 | 7 | anim2i 616 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
9 | | cpmadugsum.y |
. . . . . . 7
⊢ 𝑌 = (𝑁 Mat 𝑃) |
10 | 9 | matring 22470 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring) |
11 | 8, 10 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
12 | 11 | 3adant3 1132 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
13 | 12 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring) |
14 | | ovexd 7483 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0...𝑠) ∈ V) |
15 | | cpmadugsum.t |
. . . . . 6
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
16 | | cpmadugsum.a |
. . . . . 6
⊢ 𝐴 = (𝑁 Mat 𝑅) |
17 | | cpmadugsum.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
18 | 15, 16, 17, 5, 9 | mat2pmatbas 22753 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
19 | 4, 18 | syl3an2 1164 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
20 | 19 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
21 | 8 | 3adant3 1132 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
22 | 9 | matlmod 22456 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod) |
23 | 21, 22 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
24 | 23 | ad2antrr 725 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod) |
25 | | eqid 2740 |
. . . . . . 7
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) |
26 | | eqid 2740 |
. . . . . . 7
⊢
(Base‘𝑃) =
(Base‘𝑃) |
27 | 25, 26 | mgpbas 20167 |
. . . . . 6
⊢
(Base‘𝑃) =
(Base‘(mulGrp‘𝑃)) |
28 | | cpmadugsum.e |
. . . . . 6
⊢ ↑ =
(.g‘(mulGrp‘𝑃)) |
29 | 7 | 3ad2ant2 1134 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
30 | 25 | ringmgp 20266 |
. . . . . . . 8
⊢ (𝑃 ∈ Ring →
(mulGrp‘𝑃) ∈
Mnd) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
32 | 31 | ad2antrr 725 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd) |
33 | | elfznn0 13677 |
. . . . . . 7
⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0) |
34 | 33 | adantl 481 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ0) |
35 | 4 | 3ad2ant2 1134 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
36 | | cpmadugsum.x |
. . . . . . . . 9
⊢ 𝑋 = (var1‘𝑅) |
37 | 36, 5, 26 | vr1cl 22240 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
38 | 35, 37 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
39 | 38 | ad2antrr 725 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃)) |
40 | 27, 28, 32, 34, 39 | mulgnn0cld 19135 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
41 | 5 | ply1crng 22221 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
42 | 41 | anim2i 616 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
43 | 42 | 3adant3 1132 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
44 | 9 | matsca2 22447 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
45 | 43, 44 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌)) |
46 | 45 | eqcomd 2746 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
47 | 46 | fveq2d 6924 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
48 | 47 | eleq2d 2830 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))) |
49 | 48 | ad2antrr 725 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))) |
50 | 40, 49 | mpbird 257 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
51 | | simpll1 1212 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin) |
52 | 35 | ad2antrr 725 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring) |
53 | | simplrl 776 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0) |
54 | | simprr 772 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠))) |
55 | 54 | anim1i 614 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) |
56 | 16, 17, 5, 9, 15 | m2pmfzmap 22774 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
57 | 51, 52, 53, 55, 56 | syl31anc 1373 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
58 | | eqid 2740 |
. . . . 5
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
59 | | cpmadugsum.m |
. . . . 5
⊢ · = (
·𝑠 ‘𝑌) |
60 | | eqid 2740 |
. . . . 5
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
61 | 1, 58, 59, 60 | lmodvscl 20898 |
. . . 4
⊢ ((𝑌 ∈ LMod ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) |
62 | 24, 50, 57, 61 | syl3anc 1371 |
. . 3
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) |
63 | | simpl1 1191 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin) |
64 | 35 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring) |
65 | | simprl 770 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ0) |
66 | | eqid 2740 |
. . . . 5
⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) |
67 | | fzfid 14024 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin) |
68 | | ovexd 7483 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V) |
69 | | fvexd 6935 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) →
(0g‘𝑌)
∈ V) |
70 | 66, 67, 68, 69 | fsuppmptdm 9445 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0g‘𝑌)) |
71 | 63, 64, 65, 54, 70 | syl31anc 1373 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0g‘𝑌)) |
72 | 1, 2, 3, 13, 14, 20, 62, 71 | gsummulc2 20340 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑇‘𝑀) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) |
73 | 9 | matassa 22471 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg) |
74 | 42, 73 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg) |
75 | 74 | 3adant3 1132 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg) |
76 | 75 | ad2antrr 725 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg) |
77 | 7 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring) |
78 | 77, 30 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
(mulGrp‘𝑃) ∈
Mnd) |
79 | 78 | 3adant3 1132 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
80 | 79 | ad2antrr 725 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd) |
81 | 27, 28, 80, 34, 39 | mulgnn0cld 19135 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
82 | 47 | ad2antrr 725 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
83 | 81, 82 | eleqtrrd 2847 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
84 | 19 | ad2antrr 725 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
85 | 1, 58, 60, 59, 3 | assaassr 21902 |
. . . . 5
⊢ ((𝑌 ∈ AssAlg ∧ ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))) → ((𝑇‘𝑀) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) |
86 | 76, 83, 84, 57, 85 | syl13anc 1372 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑇‘𝑀) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) |
87 | 86 | mpteq2dva 5266 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑇‘𝑀) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) |
88 | 87 | oveq2d 7464 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑇‘𝑀) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) |
89 | 72, 88 | eqtr3d 2782 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) |