Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . . . . 7
⊢ (𝐼 Mat 𝑅) = (𝐼 Mat 𝑅) |
2 | | eqid 2622 |
. . . . . . 7
⊢
(Base‘(𝐼 Mat
𝑅)) = (Base‘(𝐼 Mat 𝑅)) |
3 | 1, 2 | matrcl 20218 |
. . . . . 6
⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 475 |
. . . . 5
⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → 𝐼 ∈ Fin) |
5 | 4 | ad3antlr 767 |
. . . 4
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin) |
6 | | isfld 18756 |
. . . . . . 7
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
7 | 6 | simplbi 476 |
. . . . . 6
⊢ (𝑅 ∈ Field → 𝑅 ∈
DivRing) |
8 | 7 | anim1i 592 |
. . . . 5
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))) |
9 | 4 | ad2antrl 764 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ Fin) |
10 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) |
11 | | xpfi 8231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐼 ∈ Fin ∧ 𝐼 ∈ Fin) → (𝐼 × 𝐼) ∈ Fin) |
12 | 11 | anidms 677 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐼 ∈ Fin → (𝐼 × 𝐼) ∈ Fin) |
13 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 freeLMod (𝐼 × 𝐼)) = (𝑅 freeLMod (𝐼 × 𝐼)) |
14 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(Base‘𝑅) =
(Base‘𝑅) |
15 | 13, 14 | frlmfibas 20105 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ DivRing ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚
(𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
16 | 12, 15 | sylan2 491 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
17 | 1, 13 | matbas 20219 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
18 | 17 | ancoms 469 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
19 | 16, 18 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅))) |
20 | 19 | eleq2d 2687 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚
(𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))) |
21 | 4, 20 | sylan2 491 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅)))) |
22 | | fvex 6201 |
. . . . . . . . . . . . . . . . . 18
⊢
(Base‘𝑅)
∈ V |
23 | 4, 4, 11 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝐼 × 𝐼) ∈ Fin) |
24 | | elmapg 7870 |
. . . . . . . . . . . . . . . . . 18
⊢
(((Base‘𝑅)
∈ V ∧ (𝐼 ×
𝐼) ∈ Fin) →
(𝑀 ∈
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
25 | 22, 23, 24 | sylancr 695 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
26 | 25 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
27 | 21, 26 | bitr3d 270 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
28 | 10, 27 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) |
29 | 28 | adantrr 753 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) |
30 | | eldifsn 4317 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 ∈ (Fin ∖ {∅})
↔ (𝐼 ∈ Fin ∧
𝐼 ≠
∅)) |
31 | 30 | biimpri 218 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 ∈ Fin ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖
{∅})) |
32 | 4, 31 | sylan 488 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → 𝐼 ∈ (Fin ∖
{∅})) |
33 | 32 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ (Fin ∖
{∅})) |
34 | | curf 33387 |
. . . . . . . . . . . . . 14
⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅}) ∧
(Base‘𝑅) ∈ V)
→ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) |
35 | 22, 34 | mp3an3 1413 |
. . . . . . . . . . . . 13
⊢ ((𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ 𝐼 ∈ (Fin ∖ {∅})) →
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) |
36 | 29, 33, 35 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) |
37 | 9, 36 | jca 554 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))) |
38 | 37 | ex 450 |
. . . . . . . . . 10
⊢ (𝑅 ∈ DivRing → ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅) → (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))) |
39 | 38 | imdistani 726 |
. . . . . . . . 9
⊢ ((𝑅 ∈ DivRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝐼 ≠ ∅)) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))) |
40 | 39 | anassrs 680 |
. . . . . . . 8
⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))) |
41 | | anass 681 |
. . . . . . . 8
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ↔ (𝑅 ∈ DivRing ∧ (𝐼 ∈ Fin ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)))) |
42 | 40, 41 | sylibr 224 |
. . . . . . 7
⊢ (((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) → ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼))) |
43 | | drngring 18754 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) |
44 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) |
45 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼) |
46 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(Base‘(𝑅
freeLMod 𝐼)) =
(Base‘(𝑅 freeLMod
𝐼)) |
47 | 44, 45, 46 | uvcff 20130 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼))) |
48 | 43, 47 | sylan 488 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 unitVec 𝐼):𝐼⟶(Base‘(𝑅 freeLMod 𝐼))) |
49 | 48 | ffvelrnda 6359 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼))) |
50 | 49 | ad4ant14 1293 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈
DivRing ∧ 𝐼 ∈ Fin)
∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ (Base‘(𝑅 freeLMod 𝐼))) |
51 | | ffn 6045 |
. . . . . . . . . . . . . . . 16
⊢ (curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → curry 𝑀 Fn 𝐼) |
52 | | fnima 6010 |
. . . . . . . . . . . . . . . 16
⊢ (curry
𝑀 Fn 𝐼 → (curry 𝑀 “ 𝐼) = ran curry 𝑀) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀 “ 𝐼) = ran curry 𝑀) |
54 | 53 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (curry 𝑀 “ 𝐼) = ran curry 𝑀) |
55 | 54 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀)) |
56 | 55 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀)) |
57 | | simplll 798 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝑅 ∈ DivRing) |
58 | | simpllr 799 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ∈ Fin) |
59 | 45 | frlmlmod 20093 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod) |
60 | 43, 59 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod) |
61 | 60 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod) |
62 | | lindfrn 20160 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) |
63 | 61, 62 | sylan 488 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) |
64 | 45 | frlmsca 20097 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
65 | | drngnzr 19262 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) |
66 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing) |
67 | 64, 66 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) →
(Scalar‘(𝑅 freeLMod
𝐼)) ∈
NzRing) |
68 | 60, 67 | jca 554 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)) |
69 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Scalar‘(𝑅
freeLMod 𝐼)) =
(Scalar‘(𝑅 freeLMod
𝐼)) |
70 | 46, 69 | lindff1 20159 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
71 | 70 | 3expa 1265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
72 | 68, 71 | sylan 488 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
73 | | fdm 6051 |
. . . . . . . . . . . . . . . . . . 19
⊢ (curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → dom curry 𝑀 = 𝐼) |
74 | | f1eq2 6097 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (dom
curry 𝑀 = 𝐼 → (curry 𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)) ↔ curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)))) |
75 | 74 | biimpac 503 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((curry
𝑀:dom curry 𝑀–1-1→(Base‘(𝑅 freeLMod 𝐼)) ∧ dom curry 𝑀 = 𝐼) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
76 | 72, 73, 75 | syl2an 494 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
77 | 76 | an32s 846 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼))) |
78 | | f1f1orn 6148 |
. . . . . . . . . . . . . . . . 17
⊢ (curry
𝑀:𝐼–1-1→(Base‘(𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1-onto→ran
curry 𝑀) |
79 | 77, 78 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → curry 𝑀:𝐼–1-1-onto→ran
curry 𝑀) |
80 | | f1oeng 7974 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∈ Fin ∧ curry 𝑀:𝐼–1-1-onto→ran
curry 𝑀) → 𝐼 ≈ ran curry 𝑀) |
81 | 58, 79, 80 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → 𝐼 ≈ ran curry 𝑀) |
82 | 81 | ensymd 8007 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ≈ 𝐼) |
83 | | lindsenlbs 33404 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ ran curry
𝑀 ∈
(LIndS‘(𝑅 freeLMod
𝐼))) ∧ ran curry 𝑀 ≈ 𝐼) → ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
84 | 57, 58, 63, 82, 83 | syl31anc 1329 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ran curry 𝑀 ∈ (LBasis‘(𝑅 freeLMod 𝐼))) |
85 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(LBasis‘(𝑅
freeLMod 𝐼)) =
(LBasis‘(𝑅 freeLMod
𝐼)) |
86 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(LSpan‘(𝑅
freeLMod 𝐼)) =
(LSpan‘(𝑅 freeLMod
𝐼)) |
87 | 46, 85, 86 | lbssp 19079 |
. . . . . . . . . . . . 13
⊢ (ran
curry 𝑀 ∈
(LBasis‘(𝑅 freeLMod
𝐼)) →
((LSpan‘(𝑅 freeLMod
𝐼))‘ran curry 𝑀) = (Base‘(𝑅 freeLMod 𝐼))) |
88 | 84, 87 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘ran curry 𝑀) = (Base‘(𝑅 freeLMod 𝐼))) |
89 | 56, 88 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))) |
90 | 89 | adantr 481 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈
DivRing ∧ 𝐼 ∈ Fin)
∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))) |
91 | 50, 90 | eleqtrrd 2704 |
. . . . . . . . 9
⊢
(((((𝑅 ∈
DivRing ∧ 𝐼 ∈ Fin)
∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼))) |
92 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) |
93 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(0g‘(Scalar‘(𝑅 freeLMod 𝐼))) =
(0g‘(Scalar‘(𝑅 freeLMod 𝐼))) |
94 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (
·𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·𝑠
‘(𝑅 freeLMod 𝐼)) |
95 | 45, 14 | frlmfibas 20105 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼))) |
96 | 95 | feq3d 6032 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ↔ curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼)))) |
97 | 96 | biimpa 501 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → curry 𝑀:𝐼⟶(Base‘(𝑅 freeLMod 𝐼))) |
98 | 59 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (𝑅 freeLMod 𝐼) ∈ LMod) |
99 | | simplr 792 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → 𝐼 ∈ Fin) |
100 | 86, 46, 92, 69, 93, 94, 97, 98, 99 | elfilspd 20142 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))) |
101 | 45 | frlmsca 20097 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼))) |
102 | 101 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
(Base‘𝑅) =
(Base‘(Scalar‘(𝑅 freeLMod 𝐼)))) |
103 | 102 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 𝐼) = ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)) |
104 | 103 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → ((Base‘𝑅) ↑𝑚
𝐼) =
((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)) |
105 | | elmapi 7879 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ((Base‘𝑅) ↑𝑚
𝐼) → 𝑛:𝐼⟶(Base‘𝑅)) |
106 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛:𝐼⟶(Base‘𝑅) → 𝑛 Fn 𝐼) |
107 | 106 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑛 Fn 𝐼) |
108 | 51 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀 Fn 𝐼) |
109 | | simpllr 799 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝐼 ∈ Fin) |
110 | | inidm 3822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
111 | | eqidd 2623 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) = (𝑛‘𝑘)) |
112 | | eqidd 2623 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) = (curry 𝑀‘𝑘)) |
113 | 107, 108,
109, 109, 110, 111, 112 | offval 6904 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)( ·𝑠
‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘)))) |
114 | | simp-4r 807 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin) |
115 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛:𝐼⟶(Base‘𝑅) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅)) |
116 | 115 | adantll 750 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅)) |
117 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
118 | 117 | ad4ant24 1298 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
119 | 95 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘(𝑅 freeLMod 𝐼))) |
120 | 118, 119 | eleqtrd 2703 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) ∈ (Base‘(𝑅 freeLMod 𝐼))) |
121 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(.r‘𝑅) = (.r‘𝑅) |
122 | 45, 46, 14, 114, 116, 120, 94, 121 | frlmvscafval 20109 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)( ·𝑠
‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘)) = ((𝐼 × {(𝑛‘𝑘)}) ∘𝑓
(.r‘𝑅)(curry 𝑀‘𝑘))) |
123 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛‘𝑘) ∈ V |
124 | | fnconstg 6093 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛‘𝑘) ∈ V → (𝐼 × {(𝑛‘𝑘)}) Fn 𝐼) |
125 | 123, 124 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝐼 × {(𝑛‘𝑘)}) Fn 𝐼) |
126 | | elmapfn 7880 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((curry
𝑀‘𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼) |
127 | 117, 126 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼) |
128 | 127 | ad4ant24 1298 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘) Fn 𝐼) |
129 | 123 | fvconst2 6469 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ 𝐼 → ((𝐼 × {(𝑛‘𝑘)})‘𝑗) = (𝑛‘𝑘)) |
130 | 129 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝐼 × {(𝑛‘𝑘)})‘𝑗) = (𝑛‘𝑘)) |
131 | | eqidd 2623 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = ((curry 𝑀‘𝑘)‘𝑗)) |
132 | 125, 128,
114, 114, 110, 130, 131 | offval 6904 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐼 × {(𝑛‘𝑘)}) ∘𝑓
(.r‘𝑅)(curry 𝑀‘𝑘)) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) |
133 | 122, 132 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)( ·𝑠
‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘)) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) |
134 | 133 | mpteq2dva 4744 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)( ·𝑠
‘(𝑅 freeLMod 𝐼))(curry 𝑀‘𝑘))) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) |
135 | 113, 134 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) |
136 | 135 | oveq2d 6666 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)) = ((𝑅 freeLMod 𝐼) Σg (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))) |
137 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢
(0g‘(𝑅 freeLMod 𝐼)) = (0g‘(𝑅 freeLMod 𝐼)) |
138 | | simplll 798 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → 𝑅 ∈ Ring) |
139 | | simp-5l 808 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring) |
140 | 115 | ad4ant23 1297 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑛‘𝑘) ∈ (Base‘𝑅)) |
141 | | simplr 792 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) |
142 | | elmapi 7879 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((curry
𝑀‘𝑘) ∈ ((Base‘𝑅) ↑𝑚 𝐼) → (curry 𝑀‘𝑘):𝐼⟶(Base‘𝑅)) |
143 | 117, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘 ∈ 𝐼) → (curry 𝑀‘𝑘):𝐼⟶(Base‘𝑅)) |
144 | 143 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((curry
𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅)) |
145 | 141, 144 | sylanl1 682 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅)) |
146 | 14, 121 | ringcl 18561 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ Ring ∧ (𝑛‘𝑘) ∈ (Base‘𝑅) ∧ ((curry 𝑀‘𝑘)‘𝑗) ∈ (Base‘𝑅)) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) ∈ (Base‘𝑅)) |
147 | 139, 140,
145, 146 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) ∈ (Base‘𝑅)) |
148 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) |
149 | 147, 148 | fmptd 6385 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅)) |
150 | | elmapg 7870 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((Base‘𝑅)
∈ V ∧ 𝐼 ∈
Fin) → ((𝑗 ∈
𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅))) |
151 | 22, 150 | mpan 706 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐼 ∈ Fin → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅))) |
152 | 151 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅))) |
153 | 95 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ ((Base‘𝑅) ↑𝑚 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼)))) |
154 | 152, 153 | bitr3d 270 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼)))) |
155 | 154 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))):𝐼⟶(Base‘𝑅) ↔ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼)))) |
156 | 149, 155 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ Fin) ∧
curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ (Base‘(𝑅 freeLMod 𝐼))) |
157 | | mptexg 6484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐼 ∈ Fin → (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V) |
158 | 157 | ralrimivw 2967 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐼 ∈ Fin → ∀𝑘 ∈ 𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V) |
159 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) = (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) |
160 | 159 | fnmpt 6020 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑘 ∈
𝐼 (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) ∈ V → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) Fn 𝐼) |
161 | 158, 160 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) Fn 𝐼) |
162 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) |
163 | | fvexd 6203 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∈ Fin →
(0g‘(𝑅
freeLMod 𝐼)) ∈
V) |
164 | 161, 162,
163 | fndmfifsupp 8288 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐼 ∈ Fin → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼))) |
165 | 164 | ad3antlr 767 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) finSupp (0g‘(𝑅 freeLMod 𝐼))) |
166 | 45, 46, 137, 109, 109, 138, 156, 165 | frlmgsum 20111 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → ((𝑅 freeLMod 𝐼) Σg (𝑘 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))) |
167 | 136, 166 | eqtr2d 2657 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛:𝐼⟶(Base‘𝑅)) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))) |
168 | 105, 167 | sylan2 491 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀))) |
169 | 168 | eqeq2d 2632 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ 𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))) |
170 | 104, 169 | rexeqbidva 3155 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (∃𝑛 ∈ ((Base‘𝑅) ↑𝑚
𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = ((𝑅 freeLMod 𝐼) Σg (𝑛 ∘𝑓 (
·𝑠 ‘(𝑅 freeLMod 𝐼))curry 𝑀)))) |
171 | 100, 170 | bitr4d 271 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))) |
172 | 43, 171 | sylanl1 682 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))) |
173 | 172 | ad2antrr 762 |
. . . . . . . . 9
⊢
(((((𝑅 ∈
DivRing ∧ 𝐼 ∈ Fin)
∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(curry 𝑀 “ 𝐼)) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))))) |
174 | 91, 173 | mpbid 222 |
. . . . . . . 8
⊢
(((((𝑅 ∈
DivRing ∧ 𝐼 ∈ Fin)
∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚
𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ 𝑖 ∈ 𝐼) → ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))) |
175 | 174 | ralrimiva 2966 |
. . . . . . 7
⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ curry 𝑀:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼)) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))) |
176 | 42, 175 | sylan 488 |
. . . . . 6
⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))))) |
177 | 10, 21 | mpbird 247 |
. . . . . . . . 9
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
178 | | elmapfn 7880 |
. . . . . . . . 9
⊢ (𝑀 ∈ ((Base‘𝑅) ↑𝑚
(𝐼 × 𝐼)) → 𝑀 Fn (𝐼 × 𝐼)) |
179 | 177, 178 | syl 17 |
. . . . . . . 8
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 Fn (𝐼 × 𝐼)) |
180 | 4 | adantl 482 |
. . . . . . . 8
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝐼 ∈ Fin) |
181 | | an32 839 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼)) |
182 | | df-3an 1039 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ↔ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼)) |
183 | 181, 182 | bitr4i 267 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ↔ (𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) |
184 | | curfv 33389 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑘 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗)) |
185 | 183, 184 | sylanb 489 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗)) |
186 | 185 | an32s 846 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((curry 𝑀‘𝑘)‘𝑗) = (𝑘𝑀𝑗)) |
187 | 186 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑘 ∈ 𝐼) → ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)) = ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) |
188 | 187 | mpteq2dva 4744 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝑗 ∈ 𝐼) ∧ 𝐼 ∈ Fin) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) |
189 | 188 | an32s 846 |
. . . . . . . . . . . . 13
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) |
190 | 189 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ (((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) ∧ 𝑗 ∈ 𝐼) → (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗)))) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
191 | 190 | mpteq2dva 4744 |
. . . . . . . . . . 11
⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) |
192 | 191 | eqeq2d 2632 |
. . . . . . . . . 10
⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
193 | 192 | rexbidv 3052 |
. . . . . . . . 9
⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∃𝑛 ∈ ((Base‘𝑅) ↑𝑚
𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
194 | 193 | ralbidv 2986 |
. . . . . . . 8
⊢ ((𝑀 Fn (𝐼 × 𝐼) ∧ 𝐼 ∈ Fin) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
195 | 179, 180,
194 | syl2anc 693 |
. . . . . . 7
⊢ ((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
196 | 195 | ad2antrr 762 |
. . . . . 6
⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → (∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)((curry 𝑀‘𝑘)‘𝑗))))) ↔ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
197 | 176, 196 | mpbid 222 |
. . . . 5
⊢ ((((𝑅 ∈ DivRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) |
198 | 8, 197 | sylanl1 682 |
. . . 4
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) |
199 | | fveq1 6190 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = ((𝑓‘𝑖)‘𝑘)) |
200 | | vex 3203 |
. . . . . . . . . . . 12
⊢ 𝑖 ∈ V |
201 | | vex 3203 |
. . . . . . . . . . . 12
⊢ 𝑘 ∈ V |
202 | | uncov 33390 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ V ∧ 𝑘 ∈ V) → (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘)) |
203 | 200, 201,
202 | mp2an 708 |
. . . . . . . . . . 11
⊢ (𝑖uncurry 𝑓𝑘) = ((𝑓‘𝑖)‘𝑘) |
204 | 199, 203 | syl6eqr 2674 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑓‘𝑖) → (𝑛‘𝑘) = (𝑖uncurry 𝑓𝑘)) |
205 | 204 | oveq1d 6665 |
. . . . . . . . 9
⊢ (𝑛 = (𝑓‘𝑖) → ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)) = ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) |
206 | 205 | mpteq2dv 4745 |
. . . . . . . 8
⊢ (𝑛 = (𝑓‘𝑖) → (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))) = (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) |
207 | 206 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑛 = (𝑓‘𝑖) → (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
208 | 207 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑛 = (𝑓‘𝑖) → (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) |
209 | 208 | eqeq2d 2632 |
. . . . 5
⊢ (𝑛 = (𝑓‘𝑖) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) ↔ ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
210 | 209 | ac6sfi 8204 |
. . . 4
⊢ ((𝐼 ∈ Fin ∧ ∀𝑖 ∈ 𝐼 ∃𝑛 ∈ ((Base‘𝑅) ↑𝑚 𝐼)((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑛‘𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
211 | 5, 198, 210 | syl2anc 693 |
. . 3
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) |
212 | | uncf 33388 |
. . . . . . 7
⊢ (𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) → uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) |
213 | 13, 14 | frlmfibas 20105 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Field ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚
(𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
214 | 12, 213 | sylan2 491 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
215 | 1, 13 | matbas 20219 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
216 | 215 | ancoms 469 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
217 | 214, 216 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅))) |
218 | 4, 217 | sylan2 491 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅))) |
219 | 218 | eleq2d 2687 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)))) |
220 | | elmapg 7870 |
. . . . . . . . . . . . . 14
⊢
(((Base‘𝑅)
∈ V ∧ (𝐼 ×
𝐼) ∈ Fin) →
(uncurry 𝑓 ∈
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
221 | 22, 23, 220 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
222 | 221 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
223 | 219, 222 | bitr3d 270 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ↔ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅))) |
224 | 223 | biimpar 502 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))) |
225 | 224 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅))) |
226 | | nfv 1843 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) |
227 | | nfmpt1 4747 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
228 | 227 | nfeq2 2780 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
229 | | fveq1 6190 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗)) |
230 | 7, 43 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 ∈ Field → 𝑅 ∈ Ring) |
231 | 230, 4 | anim12i 590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin)) |
232 | 231 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝑅 ∈ Ring ∧ 𝐼 ∈ Fin)) |
233 | | equcom 1945 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝑗 ↔ 𝑗 = 𝑖) |
234 | | ifbi 4107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 = 𝑗 ↔ 𝑗 = 𝑖) → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) |
235 | 233, 234 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)) |
236 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘𝑅) = (1r‘𝑅) |
237 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(0g‘𝑅) = (0g‘𝑅) |
238 | | simpllr 799 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin) |
239 | | simplll 798 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Ring) |
240 | | simplr 792 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼) |
241 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼) |
242 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘(𝐼 Mat 𝑅)) = (1r‘(𝐼 Mat 𝑅)) |
243 | 1, 236, 237, 238, 239, 240, 241, 242 | mat1ov 20254 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))) |
244 | | df-3an 1039 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼)) |
245 | 44, 236, 237 | uvcvval 20125 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) |
246 | 244, 245 | sylanbr 490 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) |
247 | 235, 243,
246 | 3eqtr4a 2682 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗)) |
248 | 232, 247 | sylanl1 682 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗)) |
249 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 Σg
(𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V |
250 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
251 | 250 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑗 ∈ 𝐼 ∧ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))) ∈ V) → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
252 | 249, 251 | mpan2 707 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
253 | 252 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
254 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑅 maMul 〈𝐼, 𝐼, 𝐼〉) = (𝑅 maMul 〈𝐼, 𝐼, 𝐼〉) |
255 | | simp-4l 806 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑅 ∈ Field) |
256 | 4 | ad4antlr 769 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin) |
257 | 221 | biimpar 502 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
258 | 257 | ad5ant23 1304 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → uncurry 𝑓 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
259 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) |
260 | 259, 218 | eleqtrrd 2704 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
261 | 260 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
262 | | simplr 792 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑖 ∈ 𝐼) |
263 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼) |
264 | 254, 14, 121, 255, 256, 256, 256, 258, 261, 262, 263 | mamufv 20193 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul 〈𝐼, 𝐼, 𝐼〉)𝑀)𝑗) = (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) |
265 | 1, 254 | matmulr 20244 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Field) → (𝑅 maMul 〈𝐼, 𝐼, 𝐼〉) = (.r‘(𝐼 Mat 𝑅))) |
266 | 265 | ancoms 469 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑅 maMul 〈𝐼, 𝐼, 𝐼〉) = (.r‘(𝐼 Mat 𝑅))) |
267 | 266 | oveqd 6667 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (uncurry
𝑓(𝑅 maMul 〈𝐼, 𝐼, 𝐼〉)𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)) |
268 | 267 | oveqd 6667 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅 ∈ Field ∧ 𝐼 ∈ Fin) → (𝑖(uncurry 𝑓(𝑅 maMul 〈𝐼, 𝐼, 𝐼〉)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)) |
269 | 4, 268 | sylan2 491 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑖(uncurry 𝑓(𝑅 maMul 〈𝐼, 𝐼, 𝐼〉)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)) |
270 | 269 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(𝑅 maMul 〈𝐼, 𝐼, 𝐼〉)𝑀)𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)) |
271 | 253, 264,
270 | 3eqtr2rd 2663 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗)) |
272 | 248, 271 | eqeq12d 2637 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗) ↔ (((𝑅 unitVec 𝐼)‘𝑖)‘𝑗) = ((𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))‘𝑗))) |
273 | 229, 272 | syl5ibr 236 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))) |
274 | 273 | ex 450 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))) |
275 | 274 | com23 86 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (𝑗 ∈ 𝐼 → (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗)))) |
276 | 226, 228,
275 | ralrimd 2959 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → (((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))) |
277 | 276 | ralimdva 2962 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))) |
278 | 1, 2, 242 | mat1bas 20255 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
(1r‘(𝐼 Mat
𝑅)) ∈
(Base‘(𝐼 Mat 𝑅))) |
279 | 13, 14 | frlmfibas 20105 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ Ring ∧ (𝐼 × 𝐼) ∈ Fin) → ((Base‘𝑅) ↑𝑚
(𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
280 | 12, 279 | sylan2 491 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝑅 freeLMod (𝐼 × 𝐼)))) |
281 | 1, 13 | matbas 20219 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
282 | 281 | ancoms 469 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
(Base‘(𝑅 freeLMod
(𝐼 × 𝐼))) = (Base‘(𝐼 Mat 𝑅))) |
283 | 280, 282 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅))) |
284 | 278, 283 | eleqtrrd 2704 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
(1r‘(𝐼 Mat
𝑅)) ∈
((Base‘𝑅)
↑𝑚 (𝐼 × 𝐼))) |
285 | | elmapfn 7880 |
. . . . . . . . . . . . . . . 16
⊢
((1r‘(𝐼 Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼)) |
286 | 284, 285 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) →
(1r‘(𝐼 Mat
𝑅)) Fn (𝐼 × 𝐼)) |
287 | 230, 4, 286 | syl2an 494 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼)) |
288 | 287 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼)) |
289 | 1 | matring 20249 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼 Mat 𝑅) ∈ Ring) |
290 | 4, 230, 289 | syl2anr 495 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝐼 Mat 𝑅) ∈ Ring) |
291 | 290 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (𝐼 Mat 𝑅) ∈ Ring) |
292 | | simplr 792 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) |
293 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢
(.r‘(𝐼 Mat 𝑅)) = (.r‘(𝐼 Mat 𝑅)) |
294 | 2, 293 | ringcl 18561 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐼 Mat 𝑅) ∈ Ring ∧ uncurry 𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ (Base‘(𝐼 Mat 𝑅))) |
295 | 291, 224,
292, 294 | syl3anc 1326 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ (Base‘(𝐼 Mat 𝑅))) |
296 | 218 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) = (Base‘(𝐼 Mat 𝑅))) |
297 | 295, 296 | eleqtrrd 2704 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼))) |
298 | | elmapfn 7880 |
. . . . . . . . . . . . . 14
⊢ ((uncurry
𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝐼 × 𝐼)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼)) |
299 | 297, 298 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼)) |
300 | | eqfnov2 6767 |
. . . . . . . . . . . . 13
⊢
(((1r‘(𝐼 Mat 𝑅)) Fn (𝐼 × 𝐼) ∧ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) Fn (𝐼 × 𝐼)) → ((1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ↔ ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))) |
301 | 288, 299,
300 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → ((1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) ↔ ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 (𝑖(1r‘(𝐼 Mat 𝑅))𝑗) = (𝑖(uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)𝑗))) |
302 | 277, 301 | sylibrd 249 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) → (∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀))) |
303 | 302 | imp 445 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → (1r‘(𝐼 Mat 𝑅)) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)) |
304 | 303 | eqcomd 2628 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
305 | | oveq1 6657 |
. . . . . . . . . . 11
⊢ (𝑛 = uncurry 𝑓 → (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀)) |
306 | 305 | eqeq1d 2624 |
. . . . . . . . . 10
⊢ (𝑛 = uncurry 𝑓 → ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) ↔ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) |
307 | 306 | rspcev 3309 |
. . . . . . . . 9
⊢ ((uncurry
𝑓 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (uncurry 𝑓(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
308 | 225, 304,
307 | syl2anc 693 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
309 | 308 | expl 648 |
. . . . . . 7
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((uncurry 𝑓:(𝐼 × 𝐼)⟶(Base‘𝑅) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) |
310 | 212, 309 | sylani 686 |
. . . . . 6
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) |
311 | 310 | exlimdv 1861 |
. . . . 5
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗)))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) |
312 | 311 | imp 445 |
. . . 4
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
313 | 312 | adantlr 751 |
. . 3
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ ∃𝑓(𝑓:𝐼⟶((Base‘𝑅) ↑𝑚 𝐼) ∧ ∀𝑖 ∈ 𝐼 ((𝑅 unitVec 𝐼)‘𝑖) = (𝑗 ∈ 𝐼 ↦ (𝑅 Σg (𝑘 ∈ 𝐼 ↦ ((𝑖uncurry 𝑓𝑘)(.r‘𝑅)(𝑘𝑀𝑗))))))) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
314 | 211, 313 | syldan 487 |
. 2
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ∃𝑛 ∈ (Base‘(𝐼 Mat 𝑅))(𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅))) |
315 | 6 | simprbi 480 |
. . . 4
⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing) |
316 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝐼 maDet 𝑅) = (𝐼 maDet 𝑅) |
317 | 316, 1, 2, 14 | mdetcl 20402 |
. . . . . . . . 9
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅)) |
318 | 316, 1, 2, 14 | mdetcl 20402 |
. . . . . . . . 9
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅)) |
319 | | eqid 2622 |
. . . . . . . . . 10
⊢
(∥r‘𝑅) = (∥r‘𝑅) |
320 | 14, 319, 121 | dvdsrmul 18648 |
. . . . . . . . 9
⊢ ((((𝐼 maDet 𝑅)‘𝑀) ∈ (Base‘𝑅) ∧ ((𝐼 maDet 𝑅)‘𝑛) ∈ (Base‘𝑅)) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
321 | 317, 318,
320 | syl2an 494 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
322 | 321 | anandis 873 |
. . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ (𝑀 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
323 | 322 | anassrs 680 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
324 | 323 | adantrr 753 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
325 | | fveq2 6191 |
. . . . . . . . 9
⊢ ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅)))) |
326 | 1, 2, 316, 121, 293 | mdetmul 20429 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
327 | 326 | 3expa 1265 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ CRing ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
328 | 327 | an32s 846 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀))) |
329 | 316, 1, 242, 236 | mdet1 20407 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ Fin) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅)) |
330 | 4, 329 | sylan2 491 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅)) |
331 | 330 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) = (1r‘𝑅)) |
332 | 328, 331 | eqeq12d 2637 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → (((𝐼 maDet 𝑅)‘(𝑛(.r‘(𝐼 Mat 𝑅))𝑀)) = ((𝐼 maDet 𝑅)‘(1r‘(𝐼 Mat 𝑅))) ↔ (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅))) |
333 | 325, 332 | syl5ib 234 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝑛 ∈ (Base‘(𝐼 Mat 𝑅))) → ((𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)) → (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅))) |
334 | 333 | impr 649 |
. . . . . . 7
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) = (1r‘𝑅)) |
335 | 334 | breq2d 4665 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅))) |
336 | | eqid 2622 |
. . . . . . . 8
⊢
(Unit‘𝑅) =
(Unit‘𝑅) |
337 | 336, 236,
319 | crngunit 18662 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅))) |
338 | 337 | ad2antrr 762 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅) ↔ ((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(1r‘𝑅))) |
339 | 335, 338 | bitr4d 271 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → (((𝐼 maDet 𝑅)‘𝑀)(∥r‘𝑅)(((𝐼 maDet 𝑅)‘𝑛)(.r‘𝑅)((𝐼 maDet 𝑅)‘𝑀)) ↔ ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))) |
340 | 324, 339 | mpbid 222 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)) |
341 | 315, 340 | sylanl1 682 |
. . 3
⊢ (((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)) |
342 | 341 | ad4ant14 1293 |
. 2
⊢
(((((𝑅 ∈ Field
∧ 𝑀 ∈
(Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) ∧ (𝑛 ∈ (Base‘(𝐼 Mat 𝑅)) ∧ (𝑛(.r‘(𝐼 Mat 𝑅))𝑀) = (1r‘(𝐼 Mat 𝑅)))) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)) |
343 | 314, 342 | rexlimddv 3035 |
1
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)) |