Proof of Theorem cpmadugsumlemC
Step | Hyp | Ref
| Expression |
1 | | eqid 2610 |
. . 3
⊢
(Base‘𝑌) =
(Base‘𝑌) |
2 | | eqid 2610 |
. . 3
⊢
(0g‘𝑌) = (0g‘𝑌) |
3 | | eqid 2610 |
. . 3
⊢
(+g‘𝑌) = (+g‘𝑌) |
4 | | cpmadugsum.r |
. . 3
⊢ × =
(.r‘𝑌) |
5 | | crngring 18381 |
. . . . . . . 8
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
6 | | cpmadugsum.p |
. . . . . . . . 9
⊢ 𝑃 = (Poly1‘𝑅) |
7 | 6 | ply1ring 19439 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
8 | 5, 7 | syl 17 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
9 | 8 | anim2i 591 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
10 | | cpmadugsum.y |
. . . . . . 7
⊢ 𝑌 = (𝑁 Mat 𝑃) |
11 | 10 | matring 20068 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring) |
12 | 9, 11 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
13 | 12 | 3adant3 1074 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
14 | 13 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑌 ∈ Ring) |
15 | | ovex 6577 |
. . . 4
⊢
(0...𝑠) ∈
V |
16 | 15 | a1i 11 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (0...𝑠) ∈ V) |
17 | | cpmadugsum.t |
. . . . . 6
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
18 | | cpmadugsum.a |
. . . . . 6
⊢ 𝐴 = (𝑁 Mat 𝑅) |
19 | | cpmadugsum.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
20 | 17, 18, 19, 6, 10 | mat2pmatbas 20350 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
21 | 5, 20 | syl3an2 1352 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
22 | 21 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
23 | 9 | 3adant3 1074 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
24 | 10 | matlmod 20054 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod) |
25 | 23, 24 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
26 | 25 | ad2antrr 758 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod) |
27 | 8 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
28 | | eqid 2610 |
. . . . . . . . 9
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) |
29 | 28 | ringmgp 18376 |
. . . . . . . 8
⊢ (𝑃 ∈ Ring →
(mulGrp‘𝑃) ∈
Mnd) |
30 | 27, 29 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
31 | 30 | ad2antrr 758 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd) |
32 | | elfznn0 12302 |
. . . . . . 7
⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0) |
33 | 32 | adantl 481 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ0) |
34 | 5 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
35 | | cpmadugsum.x |
. . . . . . . . 9
⊢ 𝑋 = (var1‘𝑅) |
36 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝑃) =
(Base‘𝑃) |
37 | 35, 6, 36 | vr1cl 19408 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
38 | 34, 37 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
39 | 38 | ad2antrr 758 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃)) |
40 | 28, 36 | mgpbas 18318 |
. . . . . . 7
⊢
(Base‘𝑃) =
(Base‘(mulGrp‘𝑃)) |
41 | | cpmadugsum.e |
. . . . . . 7
⊢ ↑ =
(.g‘(mulGrp‘𝑃)) |
42 | 40, 41 | mulgnn0cl 17381 |
. . . . . 6
⊢
(((mulGrp‘𝑃)
∈ Mnd ∧ 𝑖 ∈
ℕ0 ∧ 𝑋
∈ (Base‘𝑃))
→ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
43 | 31, 33, 39, 42 | syl3anc 1318 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
44 | 6 | ply1crng 19389 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
45 | 44 | anim2i 591 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
46 | 45 | 3adant3 1074 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
47 | 10 | matsca2 20045 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
48 | 46, 47 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌)) |
49 | 48 | eqcomd 2616 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
50 | 49 | fveq2d 6107 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
51 | 50 | eleq2d 2673 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))) |
52 | 51 | ad2antrr 758 |
. . . . 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))) |
53 | 43, 52 | mpbird 246 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
54 | | simpll1 1093 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin) |
55 | 34 | ad2antrr 758 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring) |
56 | | simplrl 796 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0) |
57 | | simprr 792 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) |
58 | 57 | anim1i 590 |
. . . . 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))) |
59 | 18, 19, 6, 10, 17 | m2pmfzmap 20371 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ (𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
60 | 54, 55, 56, 58, 59 | syl31anc 1321 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
61 | | eqid 2610 |
. . . . 5
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
62 | | cpmadugsum.m |
. . . . 5
⊢ · = (
·𝑠 ‘𝑌) |
63 | | eqid 2610 |
. . . . 5
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
64 | 1, 61, 62, 63 | lmodvscl 18703 |
. . . 4
⊢ ((𝑌 ∈ LMod ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) |
65 | 26, 53, 60, 64 | syl3anc 1318 |
. . 3
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) |
66 | | simpl1 1057 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑁 ∈ Fin) |
67 | 34 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑅 ∈ Ring) |
68 | | simprl 790 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑠 ∈ ℕ0) |
69 | | eqid 2610 |
. . . . 5
⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) |
70 | | fzfid 12634 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) → (0...𝑠) ∈ Fin) |
71 | | ovex 6577 |
. . . . . 6
⊢ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V |
72 | 71 | a1i 11 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V) |
73 | | fvex 6113 |
. . . . . 6
⊢
(0g‘𝑌) ∈ V |
74 | 73 | a1i 11 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) →
(0g‘𝑌)
∈ V) |
75 | 69, 70, 72, 74 | fsuppmptdm 8169 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0g‘𝑌)) |
76 | 66, 67, 68, 57, 75 | syl31anc 1321 |
. . 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)) |
77 | 1, 2, 3, 4, 14, 16, 22, 65, 76 | gsummulc2 18430 |
. 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))))))) |
78 | 10 | matassa 20069 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg) |
79 | 45, 78 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg) |
80 | 79 | 3adant3 1074 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg) |
81 | 80 | ad2antrr 758 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg) |
82 | 8 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring) |
83 | 82, 29 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
(mulGrp‘𝑃) ∈
Mnd) |
84 | 83 | 3adant3 1074 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
85 | 84 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd) |
86 | 85, 33, 39, 42 | syl3anc 1318 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
87 | 50 | ad2antrr 758 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
88 | 86, 87 | eleqtrrd 2691 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
89 | 21 | ad2antrr 758 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
90 | 1, 61, 63, 62, 4 | assaassr 19139 |
. . . . 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))))) |
91 | 81, 88, 89, 60, 90 | syl13anc 1320 |
. . . 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))))) |
92 | 91 | mpteq2dva 4672 |
. . 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)))))) |
93 | 92 | oveq2d 6565 |
. 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))))))) |
94 | 77, 93 | eqtr3d 2646 |
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))))))) |