# Metamath Proof Explorer

## Theorem minmar1eval

Description: An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018)

Ref Expression
Hypotheses minmar1fval.a 𝐴 = ( 𝑁 Mat 𝑅 )
minmar1fval.b 𝐵 = ( Base ‘ 𝐴 )
minmar1fval.q 𝑄 = ( 𝑁 minMatR1 𝑅 )
minmar1fval.o 1 = ( 1r𝑅 )
minmar1fval.z 0 = ( 0g𝑅 )
Assertion minmar1eval ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )

### Proof

Step Hyp Ref Expression
1 minmar1fval.a 𝐴 = ( 𝑁 Mat 𝑅 )
2 minmar1fval.b 𝐵 = ( Base ‘ 𝐴 )
3 minmar1fval.q 𝑄 = ( 𝑁 minMatR1 𝑅 )
4 minmar1fval.o 1 = ( 1r𝑅 )
5 minmar1fval.z 0 = ( 0g𝑅 )
6 1 2 3 4 5 minmar1val ( ( 𝑀𝐵𝐾𝑁𝐿𝑁 ) → ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
7 6 3expb ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ) → ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
8 7 3adant3 ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) → ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
9 simp3l ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) → 𝐼𝑁 )
10 simpl3r ( ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) ∧ 𝑖 = 𝐼 ) → 𝐽𝑁 )
11 4 fvexi 1 ∈ V
12 5 fvexi 0 ∈ V
13 11 12 ifex if ( 𝑗 = 𝐿 , 1 , 0 ) ∈ V
14 ovex ( 𝑖 𝑀 𝑗 ) ∈ V
15 13 14 ifex if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V
16 15 a1i ( ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) ∧ ( 𝑖 = 𝐼𝑗 = 𝐽 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
17 eqeq1 ( 𝑖 = 𝐼 → ( 𝑖 = 𝐾𝐼 = 𝐾 ) )
18 17 adantr ( ( 𝑖 = 𝐼𝑗 = 𝐽 ) → ( 𝑖 = 𝐾𝐼 = 𝐾 ) )
19 eqeq1 ( 𝑗 = 𝐽 → ( 𝑗 = 𝐿𝐽 = 𝐿 ) )
20 19 adantl ( ( 𝑖 = 𝐼𝑗 = 𝐽 ) → ( 𝑗 = 𝐿𝐽 = 𝐿 ) )
21 20 ifbid ( ( 𝑖 = 𝐼𝑗 = 𝐽 ) → if ( 𝑗 = 𝐿 , 1 , 0 ) = if ( 𝐽 = 𝐿 , 1 , 0 ) )
22 oveq12 ( ( 𝑖 = 𝐼𝑗 = 𝐽 ) → ( 𝑖 𝑀 𝑗 ) = ( 𝐼 𝑀 𝐽 ) )
23 18 21 22 ifbieq12d ( ( 𝑖 = 𝐼𝑗 = 𝐽 ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )
24 23 adantl ( ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) ∧ ( 𝑖 = 𝐼𝑗 = 𝐽 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )
25 9 10 16 24 ovmpodv2 ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) → ( ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) → ( 𝐼 ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) ) )
26 8 25 mpd ( ( 𝑀𝐵 ∧ ( 𝐾𝑁𝐿𝑁 ) ∧ ( 𝐼𝑁𝐽𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑄𝑀 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )