Step |
Hyp |
Ref |
Expression |
1 |
|
lmat22.m |
⊢ 𝑀 = ( litMat ‘ 〈“ 〈“ 𝐴 𝐵 ”〉 〈“ 𝐶 𝐷 ”〉 ”〉 ) |
2 |
|
lmat22.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
lmat22.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑉 ) |
4 |
|
lmat22.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
5 |
|
lmat22.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑉 ) |
6 |
|
lmat22det.t |
⊢ · = ( .r ‘ 𝑅 ) |
7 |
|
lmat22det.s |
⊢ − = ( -g ‘ 𝑅 ) |
8 |
|
lmat22det.v |
⊢ 𝑉 = ( Base ‘ 𝑅 ) |
9 |
|
lmat22det.j |
⊢ 𝐽 = ( ( 1 ... 2 ) maDet 𝑅 ) |
10 |
|
lmat22det.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
11 |
|
2nn |
⊢ 2 ∈ ℕ |
12 |
11
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℕ ) |
13 |
2 3
|
s2cld |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 ”〉 ∈ Word 𝑉 ) |
14 |
4 5
|
s2cld |
⊢ ( 𝜑 → 〈“ 𝐶 𝐷 ”〉 ∈ Word 𝑉 ) |
15 |
13 14
|
s2cld |
⊢ ( 𝜑 → 〈“ 〈“ 𝐴 𝐵 ”〉 〈“ 𝐶 𝐷 ”〉 ”〉 ∈ Word Word 𝑉 ) |
16 |
|
s2len |
⊢ ( ♯ ‘ 〈“ 〈“ 𝐴 𝐵 ”〉 〈“ 𝐶 𝐷 ”〉 ”〉 ) = 2 |
17 |
16
|
a1i |
⊢ ( 𝜑 → ( ♯ ‘ 〈“ 〈“ 𝐴 𝐵 ”〉 〈“ 𝐶 𝐷 ”〉 ”〉 ) = 2 ) |
18 |
1 2 3 4 5
|
lmat22lem |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 2 ) ) → ( ♯ ‘ ( 〈“ 〈“ 𝐴 𝐵 ”〉 〈“ 𝐶 𝐷 ”〉 ”〉 ‘ 𝑖 ) ) = 2 ) |
19 |
|
eqid |
⊢ ( ( 1 ... 2 ) Mat 𝑅 ) = ( ( 1 ... 2 ) Mat 𝑅 ) |
20 |
|
eqid |
⊢ ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) = ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) |
21 |
1 12 15 17 18 8 19 20 10
|
lmatcl |
⊢ ( 𝜑 → 𝑀 ∈ ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) ) |
22 |
|
2z |
⊢ 2 ∈ ℤ |
23 |
|
fzval3 |
⊢ ( 2 ∈ ℤ → ( 1 ... 2 ) = ( 1 ..^ ( 2 + 1 ) ) ) |
24 |
22 23
|
ax-mp |
⊢ ( 1 ... 2 ) = ( 1 ..^ ( 2 + 1 ) ) |
25 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
26 |
25
|
oveq2i |
⊢ ( 1 ..^ ( 2 + 1 ) ) = ( 1 ..^ 3 ) |
27 |
|
fzo13pr |
⊢ ( 1 ..^ 3 ) = { 1 , 2 } |
28 |
24 26 27
|
3eqtri |
⊢ ( 1 ... 2 ) = { 1 , 2 } |
29 |
28 9 19 20 7 6
|
m2detleib |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) ) → ( 𝐽 ‘ 𝑀 ) = ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) − ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) ) |
30 |
10 21 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑀 ) = ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) − ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) ) |
31 |
1 2 3 4 5
|
lmat22e11 |
⊢ ( 𝜑 → ( 1 𝑀 1 ) = 𝐴 ) |
32 |
1 2 3 4 5
|
lmat22e22 |
⊢ ( 𝜑 → ( 2 𝑀 2 ) = 𝐷 ) |
33 |
31 32
|
oveq12d |
⊢ ( 𝜑 → ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) = ( 𝐴 · 𝐷 ) ) |
34 |
1 2 3 4 5
|
lmat22e21 |
⊢ ( 𝜑 → ( 2 𝑀 1 ) = 𝐶 ) |
35 |
1 2 3 4 5
|
lmat22e12 |
⊢ ( 𝜑 → ( 1 𝑀 2 ) = 𝐵 ) |
36 |
34 35
|
oveq12d |
⊢ ( 𝜑 → ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) = ( 𝐶 · 𝐵 ) ) |
37 |
33 36
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) − ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) = ( ( 𝐴 · 𝐷 ) − ( 𝐶 · 𝐵 ) ) ) |
38 |
30 37
|
eqtrd |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝑀 ) = ( ( 𝐴 · 𝐷 ) − ( 𝐶 · 𝐵 ) ) ) |