Proof of Theorem cpmadugsumlemB
Step | Hyp | Ref
| Expression |
1 | | crngring 18381 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
2 | | cpmadugsum.p |
. . . . . . . . . . . . 13
⊢ 𝑃 = (Poly1‘𝑅) |
3 | 2 | ply1ring 19439 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | 1, 3 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
5 | 4 | 3ad2ant2 1076 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring) |
6 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) |
7 | 6 | ringmgp 18376 |
. . . . . . . . . 10
⊢ (𝑃 ∈ Ring →
(mulGrp‘𝑃) ∈
Mnd) |
8 | 5, 7 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd) |
9 | 8 | ad2antrr 758 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd) |
10 | | elfznn0 12302 |
. . . . . . . . 9
⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0) |
11 | 10 | adantl 481 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ0) |
12 | | 1nn0 11185 |
. . . . . . . . 9
⊢ 1 ∈
ℕ0 |
13 | 12 | a1i 11 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈
ℕ0) |
14 | 1 | 3ad2ant2 1076 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
15 | | cpmadugsum.x |
. . . . . . . . . . 11
⊢ 𝑋 = (var1‘𝑅) |
16 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝑃) =
(Base‘𝑃) |
17 | 15, 2, 16 | vr1cl 19408 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
18 | 14, 17 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
19 | 18 | ad2antrr 758 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃)) |
20 | 6, 16 | mgpbas 18318 |
. . . . . . . . 9
⊢
(Base‘𝑃) =
(Base‘(mulGrp‘𝑃)) |
21 | | cpmadugsum.e |
. . . . . . . . 9
⊢ ↑ =
(.g‘(mulGrp‘𝑃)) |
22 | | eqid 2610 |
. . . . . . . . . 10
⊢
(.r‘𝑃) = (.r‘𝑃) |
23 | 6, 22 | mgpplusg 18316 |
. . . . . . . . 9
⊢
(.r‘𝑃) = (+g‘(mulGrp‘𝑃)) |
24 | 20, 21, 23 | mulgnn0dir 17394 |
. . . . . . . 8
⊢
(((mulGrp‘𝑃)
∈ Mnd ∧ (𝑖 ∈
ℕ0 ∧ 1 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃))) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(.r‘𝑃)(1 ↑ 𝑋))) |
25 | 9, 11, 13, 19, 24 | syl13anc 1320 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(.r‘𝑃)(1 ↑ 𝑋))) |
26 | 2 | ply1crng 19389 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
27 | 26 | anim2i 591 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
28 | 27 | 3adant3 1074 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing)) |
29 | | cpmadugsum.y |
. . . . . . . . . . . 12
⊢ 𝑌 = (𝑁 Mat 𝑃) |
30 | 29 | matsca2 20045 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
31 | 28, 30 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌)) |
32 | 31 | ad2antrr 758 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 = (Scalar‘𝑌)) |
33 | 32 | fveq2d 6107 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (.r‘𝑃) =
(.r‘(Scalar‘𝑌))) |
34 | | eqidd 2611 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) = (𝑖 ↑ 𝑋)) |
35 | 20, 21 | mulg1 17371 |
. . . . . . . . . 10
⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋) |
36 | 18, 35 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ 𝑋) = 𝑋) |
37 | 36 | ad2antrr 758 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (1 ↑ 𝑋) = 𝑋) |
38 | 33, 34, 37 | oveq123d 6570 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋)(.r‘𝑃)(1 ↑ 𝑋)) = ((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋)) |
39 | 25, 38 | eqtrd 2644 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋)) |
40 | 4 | anim2i 591 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
41 | 40 | 3adant3 1074 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
42 | 29 | matring 20068 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring) |
43 | 41, 42 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
44 | 43 | ad2antrr 758 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring) |
45 | | simpll1 1093 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin) |
46 | 14 | ad2antrr 758 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring) |
47 | | simplrl 796 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0) |
48 | | simprr 792 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) |
49 | 48 | anim1i 590 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) |
50 | | cpmadugsum.a |
. . . . . . . . . 10
⊢ 𝐴 = (𝑁 Mat 𝑅) |
51 | | cpmadugsum.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐴) |
52 | | cpmadugsum.t |
. . . . . . . . . 10
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
53 | 50, 51, 2, 29, 52 | m2pmfzmap 20371 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ (𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
54 | 45, 46, 47, 49, 53 | syl31anc 1321 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) |
55 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝑌) =
(Base‘𝑌) |
56 | | cpmadugsum.r |
. . . . . . . . 9
⊢ × =
(.r‘𝑌) |
57 | | cpmadugsum.1 |
. . . . . . . . 9
⊢ 1 =
(1r‘𝑌) |
58 | 55, 56, 57 | ringlidm 18394 |
. . . . . . . 8
⊢ ((𝑌 ∈ Ring ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖))) |
59 | 44, 54, 58 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖))) |
60 | 59 | eqcomd 2616 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) = ( 1 × (𝑇‘(𝑏‘𝑖)))) |
61 | 39, 60 | oveq12d 6567 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = (((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖))))) |
62 | 29 | matassa 20069 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg) |
63 | 27, 62 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg) |
64 | 63 | 3adant3 1074 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg) |
65 | 64 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg) |
66 | 31 | eqcomd 2616 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃) |
67 | 66 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
68 | 18, 67 | eleqtrrd 2691 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌))) |
69 | 68 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘(Scalar‘𝑌))) |
70 | 20, 21 | mulgnn0cl 17381 |
. . . . . . . . 9
⊢
(((mulGrp‘𝑃)
∈ Mnd ∧ 𝑖 ∈
ℕ0 ∧ 𝑋
∈ (Base‘𝑃))
→ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
71 | 9, 11, 19, 70 | syl3anc 1318 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)) |
72 | 67 | ad2antrr 758 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
73 | 71, 72 | eleqtrrd 2691 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
74 | 40, 42 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
75 | 74 | 3adant3 1074 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
76 | 55, 57 | ringidcl 18391 |
. . . . . . . . 9
⊢ (𝑌 ∈ Ring → 1 ∈
(Base‘𝑌)) |
77 | 75, 76 | syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌)) |
78 | 77 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ (Base‘𝑌)) |
79 | | eqid 2610 |
. . . . . . . 8
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
80 | | eqid 2610 |
. . . . . . . 8
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
81 | | eqid 2610 |
. . . . . . . 8
⊢
(.r‘(Scalar‘𝑌)) =
(.r‘(Scalar‘𝑌)) |
82 | | cpmadugsum.m |
. . . . . . . 8
⊢ · = (
·𝑠 ‘𝑌) |
83 | 55, 79, 80, 81, 82, 56 | assa2ass 19143 |
. . . . . . 7
⊢ ((𝑌 ∈ AssAlg ∧ (𝑋 ∈
(Base‘(Scalar‘𝑌)) ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) ∧ ( 1 ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))) → ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖))))) |
84 | 65, 69, 73, 78, 54, 83 | syl122anc 1327 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖))))) |
85 | 84 | eqcomd 2616 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 ↑ 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖)))) = ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) |
86 | 61, 85 | eqtrd 2644 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) |
87 | 86 | mpteq2dva 4672 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) |
88 | 87 | oveq2d 6565 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) |
89 | | eqid 2610 |
. . 3
⊢
(0g‘𝑌) = (0g‘𝑌) |
90 | | eqid 2610 |
. . 3
⊢
(+g‘𝑌) = (+g‘𝑌) |
91 | 75 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑌 ∈ Ring) |
92 | | ovex 6577 |
. . . 4
⊢
(0...𝑠) ∈
V |
93 | 92 | a1i 11 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (0...𝑠) ∈ V) |
94 | 29 | matlmod 20054 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod) |
95 | 40, 94 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod) |
96 | 95 | 3adant3 1074 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod) |
97 | 1 | adantl 481 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) |
98 | 97, 17 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃)) |
99 | 27, 30 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌)) |
100 | 99 | eqcomd 2616 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
(Scalar‘𝑌) = 𝑃) |
101 | 100 | fveq2d 6107 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
(Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
102 | 98, 101 | eleqtrrd 2691 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈
(Base‘(Scalar‘𝑌))) |
103 | 102 | 3adant3 1074 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌))) |
104 | 43, 76 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌)) |
105 | 55, 79, 82, 80 | lmodvscl 18703 |
. . . . 5
⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈
(Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
106 | 96, 103, 104, 105 | syl3anc 1318 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
107 | 106 | adantr 480 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
108 | 96 | ad2antrr 758 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod) |
109 | 30 | eqcomd 2616 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) →
(Scalar‘𝑌) = 𝑃) |
110 | 109 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) →
(Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
111 | 28, 110 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃)) |
112 | 111 | eleq2d 2673 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))) |
113 | 112 | 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))) |
114 | 71, 113 | mpbird 246 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) |
115 | 55, 79, 82, 80 | lmodvscl 18703 |
. . . 4
⊢ ((𝑌 ∈ LMod ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) |
116 | 108, 114,
54, 115 | 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)) |
117 | | simpl1 1057 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑁 ∈ Fin) |
118 | 14 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑅 ∈ Ring) |
119 | | simprl 790 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑠 ∈ ℕ0) |
120 | | eqid 2610 |
. . . . 5
⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) |
121 | | fzfid 12634 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) → (0...𝑠) ∈ Fin) |
122 | | ovex 6577 |
. . . . . 6
⊢ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V |
123 | 122 | a1i 11 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V) |
124 | | fvex 6113 |
. . . . . 6
⊢
(0g‘𝑌) ∈ V |
125 | 124 | a1i 11 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0)
∧ 𝑏 ∈ (𝐵 ↑𝑚
(0...𝑠))) →
(0g‘𝑌)
∈ V) |
126 | 120, 121,
123, 125 | 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)) |
127 | 117, 118,
119, 48, 126 | 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)) |
128 | 55, 89, 90, 56, 91, 93, 107, 116, 127 | gsummulc2 18430 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) |
129 | 88, 128 | eqtr2d 2645 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) |