| Step | Hyp | Ref
| Expression |
| 1 | | cayhamlem1.a |
. . 3
⊢ 𝐴 = (𝑁 Mat 𝑅) |
| 2 | | cayhamlem1.b |
. . 3
⊢ 𝐵 = (Base‘𝐴) |
| 3 | | cayhamlem1.p |
. . 3
⊢ 𝑃 = (Poly1‘𝑅) |
| 4 | | cayhamlem1.y |
. . 3
⊢ 𝑌 = (𝑁 Mat 𝑃) |
| 5 | | cayhamlem1.r |
. . 3
⊢ × =
(.r‘𝑌) |
| 6 | | cayhamlem1.s |
. . 3
⊢ − =
(-g‘𝑌) |
| 7 | | cayhamlem1.0 |
. . 3
⊢ 0 =
(0g‘𝑌) |
| 8 | | cayhamlem1.t |
. . 3
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| 9 | | cayhamlem1.g |
. . 3
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
| 10 | | cayhamlem1.e |
. . 3
⊢ ↑ =
(.g‘(mulGrp‘𝑌)) |
| 11 | | eqid 2737 |
. . 3
⊢
(+g‘𝑌) = (+g‘𝑌) |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | chfacfpmmulgsum2 22871 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0
↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))) |
| 13 | | elfzelz 13564 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ) |
| 14 | 13 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ) |
| 15 | | pncan1 11687 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖) |
| 16 | 14, 15 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖) |
| 17 | 16 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1)) |
| 18 | 17 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1)) |
| 19 | 18 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘𝑖) = (𝑏‘((𝑖 + 1) − 1))) |
| 20 | 19 | fveq2d 6910 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1)))) |
| 21 | 20 | oveq2d 7447 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))) |
| 22 | 21 | oveq2d 7447 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))) |
| 23 | 22 | mpteq2dva 5242 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))) = (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))) |
| 24 | 23 | oveq2d 7447 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))))) |
| 25 | 24 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))))) |
| 26 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑌) =
(Base‘𝑌) |
| 27 | | crngring 20242 |
. . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
| 28 | 27 | anim2i 617 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 29 | 28 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 30 | 3, 4 | pmatring 22698 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
| 31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
| 32 | | ringabl 20278 |
. . . . . . 7
⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel) |
| 33 | 31, 32 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Abel) |
| 34 | 33 | adantr 480 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Abel) |
| 35 | | elnnuz 12922 |
. . . . . . 7
⊢ (𝑠 ∈ ℕ ↔ 𝑠 ∈
(ℤ≥‘1)) |
| 36 | 35 | biimpi 216 |
. . . . . 6
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
(ℤ≥‘1)) |
| 37 | 36 | ad2antrl 728 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈
(ℤ≥‘1)) |
| 38 | 31 | adantr 480 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring) |
| 39 | 38 | adantr 480 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑌 ∈ Ring) |
| 40 | 28, 30 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring) |
| 41 | 40 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
| 42 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(mulGrp‘𝑌) =
(mulGrp‘𝑌) |
| 43 | 42 | ringmgp 20236 |
. . . . . . . . . . . 12
⊢ (𝑌 ∈ Ring →
(mulGrp‘𝑌) ∈
Mnd) |
| 44 | 41, 43 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd) |
| 45 | 44 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd) |
| 46 | 45 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd) |
| 47 | | mndmgm 18754 |
. . . . . . . . 9
⊢
((mulGrp‘𝑌)
∈ Mnd → (mulGrp‘𝑌) ∈ Mgm) |
| 48 | 46, 47 | syl 17 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mgm) |
| 49 | | elfznn 13593 |
. . . . . . . . 9
⊢ (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ) |
| 50 | 49 | adantl 481 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑘 ∈ ℕ) |
| 51 | 8, 1, 2, 3, 4 | mat2pmatbas 22732 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 52 | 27, 51 | syl3an2 1165 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 53 | 52 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 54 | 53 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
| 55 | 42, 26 | mgpbas 20142 |
. . . . . . . . 9
⊢
(Base‘𝑌) =
(Base‘(mulGrp‘𝑌)) |
| 56 | 55, 10 | mulgnncl 19107 |
. . . . . . . 8
⊢
(((mulGrp‘𝑌)
∈ Mgm ∧ 𝑘 ∈
ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
| 57 | 48, 50, 54, 56 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
| 58 | | simpl1 1192 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin) |
| 59 | 58 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin) |
| 60 | 27 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 61 | 60 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring) |
| 62 | 61 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring) |
| 63 | | elmapi 8889 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵) |
| 64 | 63 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵) |
| 65 | 64 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵) |
| 66 | 65 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵) |
| 67 | | nnz 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℤ) |
| 68 | | peano2nn 12278 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℕ) |
| 69 | 68 | nnzd 12640 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℤ) |
| 70 | | elfzm1b 13642 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) →
(𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) −
1)))) |
| 71 | 67, 69, 70 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)))) |
| 72 | | nncn 12274 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℂ) |
| 73 | | pncan1 11687 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠) |
| 74 | 72, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠) |
| 75 | 74 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠) |
| 76 | 75 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) →
(0...((𝑠 + 1) − 1)) =
(0...𝑠)) |
| 77 | 76 | eleq2d 2827 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) ↔ (𝑘 − 1) ∈ (0...𝑠))) |
| 78 | 77 | biimpd 229 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) → (𝑘 − 1) ∈ (0...𝑠))) |
| 79 | 71, 78 | sylbid 240 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))) |
| 80 | 79 | expcom 413 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ℕ → (𝑘 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))) |
| 81 | 80 | com13 88 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 ∈ ℕ → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠)))) |
| 82 | 49, 81 | mpd 15 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠))) |
| 83 | 82 | com12 32 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))) |
| 84 | 83 | ad2antrl 728 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))) |
| 85 | 84 | imp 406 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 − 1) ∈ (0...𝑠)) |
| 86 | 66, 85 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑏‘(𝑘 − 1)) ∈ 𝐵) |
| 87 | 8, 1, 2, 3, 4 | mat2pmatbas 22732 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌)) |
| 88 | 59, 62, 86, 87 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌)) |
| 89 | 26, 5 | ringcl 20247 |
. . . . . . 7
⊢ ((𝑌 ∈ Ring ∧ (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌)) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌)) |
| 90 | 39, 57, 88, 89 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌)) |
| 91 | 90 | ralrimiva 3146 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑘 ∈ (1...(𝑠 + 1))((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌)) |
| 92 | | oveq1 7438 |
. . . . . 6
⊢ (𝑘 = 𝑖 → (𝑘 ↑ (𝑇‘𝑀)) = (𝑖 ↑ (𝑇‘𝑀))) |
| 93 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1))) |
| 94 | 93 | fveq2d 6910 |
. . . . . 6
⊢ (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1)))) |
| 95 | 92, 94 | oveq12d 7449 |
. . . . 5
⊢ (𝑘 = 𝑖 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1))))) |
| 96 | | oveq1 7438 |
. . . . . 6
⊢ (𝑘 = (𝑖 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑖 + 1) ↑ (𝑇‘𝑀))) |
| 97 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1))) |
| 98 | 97 | fveq2d 6910 |
. . . . . 6
⊢ (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1)))) |
| 99 | 96, 98 | oveq12d 7449 |
. . . . 5
⊢ (𝑘 = (𝑖 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))) |
| 100 | | oveq1 7438 |
. . . . . 6
⊢ (𝑘 = 1 → (𝑘 ↑ (𝑇‘𝑀)) = (1 ↑ (𝑇‘𝑀))) |
| 101 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1))) |
| 102 | 101 | fveq2d 6910 |
. . . . . 6
⊢ (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1)))) |
| 103 | 100, 102 | oveq12d 7449 |
. . . . 5
⊢ (𝑘 = 1 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1))))) |
| 104 | | oveq1 7438 |
. . . . . 6
⊢ (𝑘 = (𝑠 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑠 + 1) ↑ (𝑇‘𝑀))) |
| 105 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1))) |
| 106 | 105 | fveq2d 6910 |
. . . . . 6
⊢ (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1)))) |
| 107 | 104, 106 | oveq12d 7449 |
. . . . 5
⊢ (𝑘 = (𝑠 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))) |
| 108 | 26, 34, 6, 37, 91, 95, 99, 103, 107 | telgsumfz 20008 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))) = (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))) |
| 109 | 25, 108 | eqtrd 2777 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))) |
| 110 | 109 | oveq1d 7446 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) −
1)))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))) |
| 111 | 55, 10 | mulg1 19099 |
. . . . . . . 8
⊢ ((𝑇‘𝑀) ∈ (Base‘𝑌) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀)) |
| 112 | 52, 111 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀)) |
| 113 | 112 | adantr 480 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀)) |
| 114 | | 1cnd 11256 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 1 ∈
ℂ) |
| 115 | 114 | subidd 11608 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (1 − 1) =
0) |
| 116 | 115 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0)) |
| 117 | 116 | fveq2d 6910 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘(1 − 1))) = (𝑇‘(𝑏‘0))) |
| 118 | 113, 117 | oveq12d 7449 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) |
| 119 | 72 | ad2antrl 728 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℂ) |
| 120 | 119, 114 | pncand 11621 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠) |
| 121 | 120 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏‘𝑠)) |
| 122 | 121 | fveq2d 6910 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘((𝑠 + 1) − 1))) = (𝑇‘(𝑏‘𝑠))) |
| 123 | 122 | oveq2d 7447 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠)))) |
| 124 | 118, 123 | oveq12d 7449 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))) = (((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))) |
| 125 | 124 | oveq1d 7446 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) −
1)))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))) |
| 126 | | ringgrp 20235 |
. . . . . 6
⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) |
| 127 | 31, 126 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
| 128 | 127 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Grp) |
| 129 | | nnnn0 12533 |
. . . . . . . . 9
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℕ0) |
| 130 | | 0elfz 13664 |
. . . . . . . . 9
⊢ (𝑠 ∈ ℕ0
→ 0 ∈ (0...𝑠)) |
| 131 | 129, 130 | syl 17 |
. . . . . . . 8
⊢ (𝑠 ∈ ℕ → 0 ∈
(0...𝑠)) |
| 132 | 131 | ad2antrl 728 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ (0...𝑠)) |
| 133 | 65, 132 | ffvelcdmd 7105 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵) |
| 134 | 8, 1, 2, 3, 4 | mat2pmatbas 22732 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) |
| 135 | 58, 61, 133, 134 | syl3anc 1373 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) |
| 136 | 26, 5 | ringcl 20247 |
. . . . 5
⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) |
| 137 | 38, 53, 135, 136 | syl3anc 1373 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) |
| 138 | 45, 47 | syl 17 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mgm) |
| 139 | | simprl 771 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ) |
| 140 | 139 | peano2nnd 12283 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ) |
| 141 | 55, 10 | mulgnncl 19107 |
. . . . . 6
⊢
(((mulGrp‘𝑌)
∈ Mgm ∧ (𝑠 + 1)
∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
| 142 | 138, 140,
53, 141 | syl3anc 1373 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
| 143 | | nn0fz0 13665 |
. . . . . . . . 9
⊢ (𝑠 ∈ ℕ0
↔ 𝑠 ∈ (0...𝑠)) |
| 144 | 129, 143 | sylib 218 |
. . . . . . . 8
⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠)) |
| 145 | 144 | ad2antrl 728 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ (0...𝑠)) |
| 146 | 65, 145 | ffvelcdmd 7105 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘𝑠) ∈ 𝐵) |
| 147 | 8, 1, 2, 3, 4 | mat2pmatbas 22732 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑠) ∈ 𝐵) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
| 148 | 58, 61, 146, 147 | syl3anc 1373 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) |
| 149 | 26, 5 | ringcl 20247 |
. . . . 5
⊢ ((𝑌 ∈ Ring ∧ ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌)) |
| 150 | 38, 142, 148, 149 | syl3anc 1373 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌)) |
| 151 | 26, 11, 6, 7 | grpnpncan0 19054 |
. . . 4
⊢ ((𝑌 ∈ Grp ∧ (((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))) → ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 ) |
| 152 | 128, 137,
150, 151 | syl12anc 837 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 ) |
| 153 | 125, 152 | eqtrd 2777 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) −
1)))))(+g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 ) |
| 154 | 12, 110, 153 | 3eqtrd 2781 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0
↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = 0 ) |