Step |
Hyp |
Ref |
Expression |
1 |
|
marrepfval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
marrepfval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
marrepfval.q |
⊢ 𝑄 = ( 𝑁 matRRep 𝑅 ) |
4 |
|
marrepfval.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
1 2 3 4
|
marrepval0 |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑀 𝑄 𝑆 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → ( 𝑀 𝑄 𝑆 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ) |
7 |
|
simprl |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → 𝐾 ∈ 𝑁 ) |
8 |
|
simplrr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) ∧ 𝑘 = 𝐾 ) → 𝐿 ∈ 𝑁 ) |
9 |
1 2
|
matrcl |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
10 |
9
|
simpld |
⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin ) |
11 |
10 10
|
jca |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) ) |
12 |
11
|
ad3antrrr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) ) |
13 |
|
mpoexga |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ V ) |
14 |
12 13
|
syl |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ V ) |
15 |
|
eqeq2 |
⊢ ( 𝑘 = 𝐾 → ( 𝑖 = 𝑘 ↔ 𝑖 = 𝐾 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → ( 𝑖 = 𝑘 ↔ 𝑖 = 𝐾 ) ) |
17 |
|
eqeq2 |
⊢ ( 𝑙 = 𝐿 → ( 𝑗 = 𝑙 ↔ 𝑗 = 𝐿 ) ) |
18 |
17
|
ifbid |
⊢ ( 𝑙 = 𝐿 → if ( 𝑗 = 𝑙 , 𝑆 , 0 ) = if ( 𝑗 = 𝐿 , 𝑆 , 0 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → if ( 𝑗 = 𝑙 , 𝑆 , 0 ) = if ( 𝑗 = 𝐿 , 𝑆 , 0 ) ) |
20 |
16 19
|
ifbieq1d |
⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) |
21 |
20
|
mpoeq3dv |
⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) |
23 |
7 8 14 22
|
ovmpodv2 |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → ( ( 𝑀 𝑄 𝑆 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) → ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ) |
24 |
6 23
|
mpd |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) |