Step |
Hyp |
Ref |
Expression |
1 |
|
matplusgcell.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
matplusgcell.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
matplusgcell.p |
⊢ ✚ = ( +g ‘ 𝐴 ) |
4 |
|
matplusgcell.q |
⊢ + = ( +g ‘ 𝑅 ) |
5 |
1 2 3 4
|
matplusg2 |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ✚ 𝑌 ) = ( 𝑋 ∘f + 𝑌 ) ) |
6 |
5
|
oveqd |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ( 𝑋 ✚ 𝑌 ) 𝐽 ) = ( 𝐼 ( 𝑋 ∘f + 𝑌 ) 𝐽 ) ) |
7 |
6
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝑋 ✚ 𝑌 ) 𝐽 ) = ( 𝐼 ( 𝑋 ∘f + 𝑌 ) 𝐽 ) ) |
8 |
|
df-ov |
⊢ ( 𝐼 ( 𝑋 ∘f + 𝑌 ) 𝐽 ) = ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝐼 , 𝐽 〉 ) |
9 |
8
|
a1i |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝑋 ∘f + 𝑌 ) 𝐽 ) = ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝐼 , 𝐽 〉 ) ) |
10 |
|
opelxp |
⊢ ( 〈 𝐼 , 𝐽 〉 ∈ ( 𝑁 × 𝑁 ) ↔ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
12 |
1 11 2
|
matbas2i |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
13 |
|
elmapfn |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) → 𝑋 Fn ( 𝑁 × 𝑁 ) ) |
14 |
12 13
|
syl |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 Fn ( 𝑁 × 𝑁 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 Fn ( 𝑁 × 𝑁 ) ) |
16 |
1 11 2
|
matbas2i |
⊢ ( 𝑌 ∈ 𝐵 → 𝑌 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
17 |
|
elmapfn |
⊢ ( 𝑌 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) → 𝑌 Fn ( 𝑁 × 𝑁 ) ) |
18 |
16 17
|
syl |
⊢ ( 𝑌 ∈ 𝐵 → 𝑌 Fn ( 𝑁 × 𝑁 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 Fn ( 𝑁 × 𝑁 ) ) |
20 |
1 2
|
matrcl |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
21 |
|
xpfi |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑁 × 𝑁 ) ∈ Fin ) |
22 |
21
|
anidms |
⊢ ( 𝑁 ∈ Fin → ( 𝑁 × 𝑁 ) ∈ Fin ) |
23 |
22
|
adantr |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( 𝑁 × 𝑁 ) ∈ Fin ) |
24 |
20 23
|
syl |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝑁 × 𝑁 ) ∈ Fin ) |
25 |
24
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 × 𝑁 ) ∈ Fin ) |
26 |
|
inidm |
⊢ ( ( 𝑁 × 𝑁 ) ∩ ( 𝑁 × 𝑁 ) ) = ( 𝑁 × 𝑁 ) |
27 |
|
df-ov |
⊢ ( 𝐼 𝑋 𝐽 ) = ( 𝑋 ‘ 〈 𝐼 , 𝐽 〉 ) |
28 |
27
|
eqcomi |
⊢ ( 𝑋 ‘ 〈 𝐼 , 𝐽 〉 ) = ( 𝐼 𝑋 𝐽 ) |
29 |
28
|
a1i |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 〈 𝐼 , 𝐽 〉 ∈ ( 𝑁 × 𝑁 ) ) → ( 𝑋 ‘ 〈 𝐼 , 𝐽 〉 ) = ( 𝐼 𝑋 𝐽 ) ) |
30 |
|
df-ov |
⊢ ( 𝐼 𝑌 𝐽 ) = ( 𝑌 ‘ 〈 𝐼 , 𝐽 〉 ) |
31 |
30
|
eqcomi |
⊢ ( 𝑌 ‘ 〈 𝐼 , 𝐽 〉 ) = ( 𝐼 𝑌 𝐽 ) |
32 |
31
|
a1i |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 〈 𝐼 , 𝐽 〉 ∈ ( 𝑁 × 𝑁 ) ) → ( 𝑌 ‘ 〈 𝐼 , 𝐽 〉 ) = ( 𝐼 𝑌 𝐽 ) ) |
33 |
15 19 25 25 26 29 32
|
ofval |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 〈 𝐼 , 𝐽 〉 ∈ ( 𝑁 × 𝑁 ) ) → ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝐼 , 𝐽 〉 ) = ( ( 𝐼 𝑋 𝐽 ) + ( 𝐼 𝑌 𝐽 ) ) ) |
34 |
10 33
|
sylan2br |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝐼 , 𝐽 〉 ) = ( ( 𝐼 𝑋 𝐽 ) + ( 𝐼 𝑌 𝐽 ) ) ) |
35 |
7 9 34
|
3eqtrd |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝑋 ✚ 𝑌 ) 𝐽 ) = ( ( 𝐼 𝑋 𝐽 ) + ( 𝐼 𝑌 𝐽 ) ) ) |