Step | Hyp | Ref
| Expression |
1 | | mdetdiaglem.z |
. . . . . 6
⊢ 𝑍 = (ℤRHom‘𝑅) |
2 | 1 | a1i 11 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → 𝑍 = (ℤRHom‘𝑅)) |
3 | | mdetdiaglem.s |
. . . . . 6
⊢ 𝑆 = (pmSgn‘𝑁) |
4 | 3 | a1i 11 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → 𝑆 = (pmSgn‘𝑁)) |
5 | 2, 4 | coeq12d 5286 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (𝑍 ∘ 𝑆) = ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))) |
6 | 5 | fveq1d 6193 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((𝑍 ∘ 𝑆)‘𝑃) = (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃)) |
7 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(SymGrp‘𝑁) =
(SymGrp‘𝑁) |
8 | | mdetdiaglem.g |
. . . . . . . . . . . 12
⊢ 𝐻 =
(Base‘(SymGrp‘𝑁)) |
9 | 7, 8 | symgbasf1o 17803 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁–1-1-onto→𝑁) |
10 | | f1ofn 6138 |
. . . . . . . . . . 11
⊢ (𝑃:𝑁–1-1-onto→𝑁 → 𝑃 Fn 𝑁) |
11 | 9, 10 | syl 17 |
. . . . . . . . . 10
⊢ (𝑃 ∈ 𝐻 → 𝑃 Fn 𝑁) |
12 | | fnnfpeq0 6444 |
. . . . . . . . . 10
⊢ (𝑃 Fn 𝑁 → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁))) |
13 | 11, 12 | syl 17 |
. . . . . . . . 9
⊢ (𝑃 ∈ 𝐻 → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁))) |
14 | 13 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (dom (𝑃 ∖ I ) = ∅ ↔ 𝑃 = ( I ↾ 𝑁))) |
15 | 14 | bicomd 213 |
. . . . . . 7
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 = ( I ↾ 𝑁) ↔ dom (𝑃 ∖ I ) = ∅)) |
16 | 15 | necon3bid 2838 |
. . . . . 6
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 ≠ ( I ↾ 𝑁) ↔ dom (𝑃 ∖ I ) ≠ ∅)) |
17 | | n0 3931 |
. . . . . . 7
⊢ (dom
(𝑃 ∖ I ) ≠ ∅
↔ ∃𝑠 𝑠 ∈ dom (𝑃 ∖ I )) |
18 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(Base‘𝐺) =
(Base‘𝐺) |
19 | | mdetdiag.g |
. . . . . . . . . . . 12
⊢ 𝐺 = (mulGrp‘𝑅) |
20 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(.r‘𝑅) = (.r‘𝑅) |
21 | 19, 20 | mgpplusg 18493 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (+g‘𝐺) |
22 | 19 | crngmgp 18555 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
23 | 22 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ CMnd) |
24 | 23 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝐺 ∈ CMnd) |
25 | | simpll2 1101 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑁 ∈ Fin) |
26 | | mdetdiag.a |
. . . . . . . . . . . . . . . . 17
⊢ 𝐴 = (𝑁 Mat 𝑅) |
27 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘𝑅) =
(Base‘𝑅) |
28 | | mdetdiag.b |
. . . . . . . . . . . . . . . . 17
⊢ 𝐵 = (Base‘𝐴) |
29 | 26, 27, 28 | matbas2i 20228 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
30 | 29 | 3ad2ant3 1084 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁))) |
31 | | elmapi 7879 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ((Base‘𝑅) ↑𝑚
(𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
33 | 19, 27 | mgpbas 18495 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘𝑅) =
(Base‘𝐺) |
34 | 33 | eqcomi 2631 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝐺) =
(Base‘𝑅) |
35 | 34 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (Base‘𝐺) = (Base‘𝑅)) |
36 | 35 | feq3d 6032 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺) ↔ 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))) |
37 | 32, 36 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺)) |
38 | 37 | ad3antrrr 766 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ CRing
∧ 𝑁 ∈ Fin ∧
𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝐺)) |
39 | 7, 8 | symgbasf 17804 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈ 𝐻 → 𝑃:𝑁⟶𝑁) |
40 | 39 | ad2antrl 764 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁) |
41 | 40 | ffvelrnda 6359 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ CRing
∧ 𝑁 ∈ Fin ∧
𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → (𝑃‘𝑘) ∈ 𝑁) |
42 | | simpr 477 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ CRing
∧ 𝑁 ∈ Fin ∧
𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁) |
43 | 38, 41, 42 | fovrnd 6806 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈ CRing
∧ 𝑁 ∈ Fin ∧
𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) ∧ 𝑘 ∈ 𝑁) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝐺)) |
44 | | disjdif 4040 |
. . . . . . . . . . . 12
⊢ ({𝑠} ∩ (𝑁 ∖ {𝑠})) = ∅ |
45 | 44 | a1i 11 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ({𝑠} ∩ (𝑁 ∖ {𝑠})) = ∅) |
46 | | difss 3737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 ∖ I ) ⊆ 𝑃 |
47 | | dmss 5323 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃) |
48 | 46, 47 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ dom
(𝑃 ∖ I ) ⊆ dom
𝑃 |
49 | 39 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁) |
50 | | fdm 6051 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃:𝑁⟶𝑁 → dom 𝑃 = 𝑁) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom 𝑃 = 𝑁) |
52 | 48, 51 | syl5sseq 3653 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁) |
53 | 52 | sseld 3602 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁)) |
54 | 53 | impr 649 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁) |
55 | 54 | snssd 4340 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → {𝑠} ⊆ 𝑁) |
56 | | undif 4049 |
. . . . . . . . . . . . 13
⊢ ({𝑠} ⊆ 𝑁 ↔ ({𝑠} ∪ (𝑁 ∖ {𝑠})) = 𝑁) |
57 | 55, 56 | sylib 208 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ({𝑠} ∪ (𝑁 ∖ {𝑠})) = 𝑁) |
58 | 57 | eqcomd 2628 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑁 = ({𝑠} ∪ (𝑁 ∖ {𝑠}))) |
59 | 18, 21, 24, 25, 43, 45, 58 | gsummptfidmsplit 18330 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝐺 Σg (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘)))(.r‘𝑅)(𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))))) |
60 | | crngring 18558 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
61 | 60 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring) |
62 | 19 | ringmgp 18553 |
. . . . . . . . . . . . . . . 16
⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
63 | 61, 62 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐺 ∈ Mnd) |
64 | 63 | 3adant3 1081 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
65 | 64 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝐺 ∈ Mnd) |
66 | | vex 3203 |
. . . . . . . . . . . . . 14
⊢ 𝑠 ∈ V |
67 | 66 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ V) |
68 | 32 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
69 | 40, 54 | ffvelrnd 6360 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁) |
70 | 68, 69, 54 | fovrnd 6806 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠)𝑀𝑠) ∈ (Base‘𝑅)) |
71 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑠 → (𝑃‘𝑘) = (𝑃‘𝑠)) |
72 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑠 → 𝑘 = 𝑠) |
73 | 71, 72 | oveq12d 6668 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑠 → ((𝑃‘𝑘)𝑀𝑘) = ((𝑃‘𝑠)𝑀𝑠)) |
74 | 33, 73 | gsumsn 18354 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∈ Mnd ∧ 𝑠 ∈ V ∧ ((𝑃‘𝑠)𝑀𝑠) ∈ (Base‘𝑅)) → (𝐺 Σg (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝑃‘𝑠)𝑀𝑠)) |
75 | 65, 67, 70, 74 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σg (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = ((𝑃‘𝑠)𝑀𝑠)) |
76 | | simprr 796 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ dom (𝑃 ∖ I )) |
77 | 11 | ad2antrl 764 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃 Fn 𝑁) |
78 | | fnelnfp 6443 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 Fn 𝑁 ∧ 𝑠 ∈ 𝑁) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠)) |
79 | 77, 54, 78 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑠 ∈ dom (𝑃 ∖ I ) ↔ (𝑃‘𝑠) ≠ 𝑠)) |
80 | 76, 79 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ≠ 𝑠) |
81 | 39 | ad2antrl 764 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑃:𝑁⟶𝑁) |
82 | 39 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑃:𝑁⟶𝑁) |
83 | 82, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom 𝑃 = 𝑁) |
84 | 48, 83 | syl5sseq 3653 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → dom (𝑃 ∖ I ) ⊆ 𝑁) |
85 | 84 | sseld 3602 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → 𝑠 ∈ 𝑁)) |
86 | 85 | impr 649 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑠 ∈ 𝑁) |
87 | 81, 86 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝑃‘𝑠) ∈ 𝑁) |
88 | | neeq1 2856 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑃‘𝑠) → (𝑖 ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑗)) |
89 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = (𝑃‘𝑠) → (𝑖𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑗)) |
90 | 89 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑗) = 0 )) |
91 | 88, 90 | imbi12d 334 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑃‘𝑠) → ((𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 ))) |
92 | | neeq2 2857 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠) ≠ 𝑗 ↔ (𝑃‘𝑠) ≠ 𝑠)) |
93 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑠 → ((𝑃‘𝑠)𝑀𝑗) = ((𝑃‘𝑠)𝑀𝑠)) |
94 | 93 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠)𝑀𝑗) = 0 ↔ ((𝑃‘𝑠)𝑀𝑠) = 0 )) |
95 | 92, 94 | imbi12d 334 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑠 → (((𝑃‘𝑠) ≠ 𝑗 → ((𝑃‘𝑠)𝑀𝑗) = 0 ) ↔ ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 ))) |
96 | 91, 95 | rspc2v 3322 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑃‘𝑠) ∈ 𝑁 ∧ 𝑠 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 ))) |
97 | 87, 86, 96 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 ))) |
98 | 97 | impancom 456 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → ((𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I )) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 ))) |
99 | 98 | imp 445 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠) ≠ 𝑠 → ((𝑃‘𝑠)𝑀𝑠) = 0 )) |
100 | 80, 99 | mpd 15 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝑃‘𝑠)𝑀𝑠) = 0 ) |
101 | 75, 100 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σg (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ) |
102 | 101 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ((𝐺 Σg (𝑘 ∈ {𝑠} ↦ ((𝑃‘𝑘)𝑀𝑘)))(.r‘𝑅)(𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = ( 0 (.r‘𝑅)(𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))))) |
103 | 60 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
104 | 103 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → 𝑅 ∈ Ring) |
105 | 23 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝐺 ∈ CMnd) |
106 | | simpl2 1065 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → 𝑁 ∈ Fin) |
107 | | difss 3737 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∖ {𝑠}) ⊆ 𝑁 |
108 | | ssfi 8180 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝑠}) ⊆ 𝑁) → (𝑁 ∖ {𝑠}) ∈ Fin) |
109 | 106, 107,
108 | sylancl 694 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝑁 ∖ {𝑠}) ∈ Fin) |
110 | 32 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
111 | 82 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑃:𝑁⟶𝑁) |
112 | | eldifi 3732 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (𝑁 ∖ {𝑠}) → 𝑘 ∈ 𝑁) |
113 | 112 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → 𝑘 ∈ 𝑁) |
114 | 111, 113 | ffvelrnd 6360 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → (𝑃‘𝑘) ∈ 𝑁) |
115 | 110, 114,
113 | fovrnd 6806 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) ∧ 𝑘 ∈ (𝑁 ∖ {𝑠})) → ((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅)) |
116 | 115 | ralrimiva 2966 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → ∀𝑘 ∈ (𝑁 ∖ {𝑠})((𝑃‘𝑘)𝑀𝑘) ∈ (Base‘𝑅)) |
117 | 33, 105, 109, 116 | gsummptcl 18366 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ 𝑃 ∈ 𝐻) → (𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅)) |
118 | 117 | ad2ant2r 783 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅)) |
119 | | mdetdiag.0 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘𝑅) |
120 | 27, 20, 119 | ringlz 18587 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝐺 Σg
(𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘))) ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅)(𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 ) |
121 | 104, 118,
120 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → ( 0 (.r‘𝑅)(𝐺 Σg (𝑘 ∈ (𝑁 ∖ {𝑠}) ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 ) |
122 | 59, 102, 121 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ (𝑃 ∈ 𝐻 ∧ 𝑠 ∈ dom (𝑃 ∖ I ))) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ) |
123 | 122 | expr 643 |
. . . . . . . 8
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑠 ∈ dom (𝑃 ∖ I ) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )) |
124 | 123 | exlimdv 1861 |
. . . . . . 7
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (∃𝑠 𝑠 ∈ dom (𝑃 ∖ I ) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )) |
125 | 17, 124 | syl5bi 232 |
. . . . . 6
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (dom (𝑃 ∖ I ) ≠ ∅ → (𝐺 Σg
(𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )) |
126 | 16, 125 | sylbid 230 |
. . . . 5
⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) ∧ 𝑃 ∈ 𝐻) → (𝑃 ≠ ( I ↾ 𝑁) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )) |
127 | 126 | expimpd 629 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 )) |
128 | 127 | 3impia 1261 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘))) = 0 ) |
129 | 6, 128 | oveq12d 6668 |
. 2
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 )) |
130 | | 3simpa 1058 |
. . . 4
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ CRing ∧ 𝑁 ∈ Fin)) |
131 | | simpl 473 |
. . . 4
⊢ ((𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁)) → 𝑃 ∈ 𝐻) |
132 | 60 | ad2antrr 762 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → 𝑅 ∈ Ring) |
133 | | zrhpsgnmhm 19930 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) →
((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))
∈ ((SymGrp‘𝑁)
MndHom (mulGrp‘𝑅))) |
134 | 60, 133 | sylan 488 |
. . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) →
((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))
∈ ((SymGrp‘𝑁)
MndHom (mulGrp‘𝑅))) |
135 | | eqid 2622 |
. . . . . . . 8
⊢
(Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) |
136 | 8, 135 | mhmf 17340 |
. . . . . . 7
⊢
(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅))) |
137 | 134, 136 | syl 17 |
. . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) →
((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁)):𝐻⟶(Base‘(mulGrp‘𝑅))) |
138 | 137 | ffvelrnda 6359 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) |
139 | | eqid 2622 |
. . . . . . . 8
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
140 | 139, 27 | mgpbas 18495 |
. . . . . . 7
⊢
(Base‘𝑅) =
(Base‘(mulGrp‘𝑅)) |
141 | 140 | eqcomi 2631 |
. . . . . 6
⊢
(Base‘(mulGrp‘𝑅)) = (Base‘𝑅) |
142 | | mdetdiaglem.t |
. . . . . 6
⊢ · =
(.r‘𝑅) |
143 | 141, 142,
119 | ringrz 18588 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧
(((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝑃) ∈ (Base‘(mulGrp‘𝑅))) →
((((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 ) |
144 | 132, 138,
143 | syl2anc 693 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ 𝑃 ∈ 𝐻) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 ) |
145 | 130, 131,
144 | syl2an 494 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 ) |
146 | 145 | 3adant2 1080 |
. 2
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑃) · 0 ) = 0 ) |
147 | 129, 146 | eqtrd 2656 |
1
⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 ) |