| Step | Hyp | Ref
| Expression |
| 1 | | simpll 767 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin) |
| 2 | | simplr 769 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing) |
| 3 | | elrabi 3687 |
. . . . . 6
⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} → 𝑀 ∈ (Base‘𝐴)) |
| 4 | | chpscmat.d |
. . . . . 6
⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} |
| 5 | 3, 4 | eleq2s 2859 |
. . . . 5
⊢ (𝑀 ∈ 𝐷 → 𝑀 ∈ (Base‘𝐴)) |
| 6 | 5 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → 𝑀 ∈ (Base‘𝐴)) |
| 7 | 6 | adantl 481 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑀 ∈ (Base‘𝐴)) |
| 8 | | oveq 7437 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗)) |
| 9 | 8 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)))) |
| 10 | 9 | 2ralbidv 3221 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)))) |
| 11 | 10 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)))) |
| 12 | 11 | elrab 3692 |
. . . . . . 7
⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} ↔ (𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)))) |
| 13 | | ifnefalse 4537 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ≠ 𝑗 → if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) = (0g‘𝑅)) |
| 14 | 13 | eqeq2d 2748 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ≠ 𝑗 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = (0g‘𝑅))) |
| 15 | 14 | biimpcd 249 |
. . . . . . . . . . . . . 14
⊢ ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))) |
| 16 | 15 | a1i 11 |
. . . . . . . . . . . . 13
⊢
(((((𝑀 ∈
(Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 17 | 16 | ralimdva 3167 |
. . . . . . . . . . . 12
⊢ ((((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖 ∈ 𝑁) → (∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 18 | 17 | ralimdva 3167 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 19 | 18 | ex 412 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))) |
| 20 | 19 | com23 86 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))) |
| 21 | 20 | rexlimdva 3155 |
. . . . . . . 8
⊢ (𝑀 ∈ (Base‘𝐴) → (∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))) |
| 22 | 21 | imp 406 |
. . . . . . 7
⊢ ((𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 23 | 12, 22 | sylbi 217 |
. . . . . 6
⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 24 | 23, 4 | eleq2s 2859 |
. . . . 5
⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 25 | 24 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))) |
| 26 | 25 | impcom 407 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))) |
| 27 | | chp0mat.c |
. . . 4
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
| 28 | | chp0mat.p |
. . . 4
⊢ 𝑃 = (Poly1‘𝑅) |
| 29 | | chp0mat.a |
. . . 4
⊢ 𝐴 = (𝑁 Mat 𝑅) |
| 30 | | chpscmat.s |
. . . 4
⊢ 𝑆 = (algSc‘𝑃) |
| 31 | | eqid 2737 |
. . . 4
⊢
(Base‘𝐴) =
(Base‘𝐴) |
| 32 | | chp0mat.x |
. . . 4
⊢ 𝑋 = (var1‘𝑅) |
| 33 | | eqid 2737 |
. . . 4
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 34 | | chp0mat.g |
. . . 4
⊢ 𝐺 = (mulGrp‘𝑃) |
| 35 | | chpscmat.m |
. . . 4
⊢ − =
(-g‘𝑃) |
| 36 | 27, 28, 29, 30, 31, 32, 33, 34, 35 | chpdmat 22847 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘𝐴)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘)))))) |
| 37 | 1, 2, 7, 26, 36 | syl31anc 1375 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘)))))) |
| 38 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) |
| 39 | 38, 38 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘)) |
| 40 | 39 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸)) |
| 41 | 40 | rspccv 3619 |
. . . . . . . . 9
⊢
(∀𝑛 ∈
𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸)) |
| 42 | 41 | 3ad2ant3 1136 |
. . . . . . . 8
⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸)) |
| 43 | 42 | adantl 481 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸)) |
| 44 | 43 | imp 406 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑀𝑘) = 𝐸) |
| 45 | 44 | fveq2d 6910 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑆‘(𝑘𝑀𝑘)) = (𝑆‘𝐸)) |
| 46 | 45 | oveq2d 7447 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑋 − (𝑆‘(𝑘𝑀𝑘))) = (𝑋 − (𝑆‘𝐸))) |
| 47 | 46 | mpteq2dva 5242 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘)))) = (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))) |
| 48 | 47 | oveq2d 7447 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸))))) |
| 49 | 28 | ply1crng 22200 |
. . . . 5
⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 50 | 34 | crngmgp 20238 |
. . . . 5
⊢ (𝑃 ∈ CRing → 𝐺 ∈ CMnd) |
| 51 | | cmnmnd 19815 |
. . . . 5
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
| 52 | 49, 50, 51 | 3syl 18 |
. . . 4
⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd) |
| 53 | 52 | ad2antlr 727 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd) |
| 54 | | crngring 20242 |
. . . . . . . 8
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
| 55 | 28 | ply1ring 22249 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
| 56 | 54, 55 | syl 17 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
| 57 | | ringgrp 20235 |
. . . . . . 7
⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) |
| 58 | 56, 57 | syl 17 |
. . . . . 6
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp) |
| 59 | 58 | ad2antlr 727 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp) |
| 60 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝑃) =
(Base‘𝑃) |
| 61 | 32, 28, 60 | vr1cl 22219 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
| 62 | 54, 61 | syl 17 |
. . . . . 6
⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) |
| 63 | 62 | ad2antlr 727 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑋 ∈ (Base‘𝑃)) |
| 64 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝐼 ∈ 𝑁) |
| 65 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
| 66 | 56 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ Ring) |
| 67 | 66 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ Ring) |
| 68 | 28 | ply1lmod 22253 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
| 69 | 54, 68 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑅 ∈ CRing → 𝑃 ∈ LMod) |
| 70 | 69 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ LMod) |
| 71 | 70 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ LMod) |
| 72 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
| 73 | 30, 65, 67, 71, 72, 60 | asclf 21902 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑆:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃)) |
| 74 | 5 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑀 ∈ (Base‘𝐴)) |
| 75 | 74 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴)) |
| 76 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 77 | 29, 76 | matecl 22431 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅)) |
| 78 | 64, 64, 75, 77 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅)) |
| 79 | 28 | ply1sca 22254 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃)) |
| 80 | 79 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑅 = (Scalar‘𝑃)) |
| 81 | 80 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃)) |
| 82 | 81 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅) |
| 83 | 82 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅)) |
| 84 | 78, 83 | eleqtrrd 2844 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃))) |
| 85 | 73, 84 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃)) |
| 86 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐸 = (𝐼𝑀𝐼) → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼))) |
| 87 | 86 | eqcoms 2745 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼))) |
| 88 | 87 | eleq1d 2826 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼𝑀𝐼) = 𝐸 → ((𝑆‘𝐸) ∈ (Base‘𝑃) ↔ (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃))) |
| 89 | 85, 88 | syl5ibrcom 247 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))) |
| 90 | 89 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))) |
| 91 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝐼 → 𝑛 = 𝐼) |
| 92 | 91, 91 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼)) |
| 93 | 92 | eqeq1d 2739 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝐼 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝐼𝑀𝐼) = 𝐸)) |
| 94 | 93 | imbi1d 341 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝐼 → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))) |
| 95 | 94 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))) |
| 96 | 90, 95 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))) |
| 97 | 64, 96 | rspcimdv 3612 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))) |
| 98 | 97 | ex 412 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))) |
| 99 | 98 | com23 86 |
. . . . . . . . 9
⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃)))) |
| 100 | 99 | ex 412 |
. . . . . . . 8
⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃))))) |
| 101 | 100 | com24 95 |
. . . . . . 7
⊢ (𝑀 ∈ 𝐷 → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃))))) |
| 102 | 101 | 3imp 1111 |
. . . . . 6
⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃))) |
| 103 | 102 | impcom 407 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑆‘𝐸) ∈ (Base‘𝑃)) |
| 104 | 60, 35 | grpsubcl 19038 |
. . . . 5
⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆‘𝐸) ∈ (Base‘𝑃)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃)) |
| 105 | 59, 63, 103, 104 | syl3anc 1373 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃)) |
| 106 | 34, 60 | mgpbas 20142 |
. . . 4
⊢
(Base‘𝑃) =
(Base‘𝐺) |
| 107 | 105, 106 | eleqtrdi 2851 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺)) |
| 108 | | eqid 2737 |
. . . 4
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 109 | | chp0mat.m |
. . . 4
⊢ ↑ =
(.g‘𝐺) |
| 110 | 108, 109 | gsumconst 19952 |
. . 3
⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ Fin ∧ (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸)))) |
| 111 | 53, 1, 107, 110 | syl3anc 1373 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸)))) |
| 112 | 37, 48, 111 | 3eqtrd 2781 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸)))) |