Step |
Hyp |
Ref |
Expression |
1 |
|
mamumat1cl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
mamumat1cl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
3 |
|
mamumat1cl.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
4 |
|
mamumat1cl.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
mamumat1cl.i |
⊢ 𝐼 = ( 𝑖 ∈ 𝑀 , 𝑗 ∈ 𝑀 ↦ if ( 𝑖 = 𝑗 , 1 , 0 ) ) |
6 |
|
mamumat1cl.m |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
7 |
1 3
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 ) |
8 |
1 4
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 ) |
9 |
7 8
|
ifcld |
⊢ ( 𝑅 ∈ Ring → if ( 𝑖 = 𝑗 , 1 , 0 ) ∈ 𝐵 ) |
10 |
2 9
|
syl |
⊢ ( 𝜑 → if ( 𝑖 = 𝑗 , 1 , 0 ) ∈ 𝐵 ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀 ) ) → if ( 𝑖 = 𝑗 , 1 , 0 ) ∈ 𝐵 ) |
12 |
11
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑀 ∀ 𝑗 ∈ 𝑀 if ( 𝑖 = 𝑗 , 1 , 0 ) ∈ 𝐵 ) |
13 |
5
|
fmpo |
⊢ ( ∀ 𝑖 ∈ 𝑀 ∀ 𝑗 ∈ 𝑀 if ( 𝑖 = 𝑗 , 1 , 0 ) ∈ 𝐵 ↔ 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) |
14 |
12 13
|
sylib |
⊢ ( 𝜑 → 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) |
15 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
16 |
|
xpfi |
⊢ ( ( 𝑀 ∈ Fin ∧ 𝑀 ∈ Fin ) → ( 𝑀 × 𝑀 ) ∈ Fin ) |
17 |
6 6 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 × 𝑀 ) ∈ Fin ) |
18 |
|
elmapg |
⊢ ( ( 𝐵 ∈ V ∧ ( 𝑀 × 𝑀 ) ∈ Fin ) → ( 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) ↔ 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) ) |
19 |
15 17 18
|
sylancr |
⊢ ( 𝜑 → ( 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) ↔ 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) ) |
20 |
14 19
|
mpbird |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) ) |