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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 = ( 1r ‘ 𝑅 )
minmar1fval.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion minmar1eval ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑄 ‘ 𝑀 ) 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 = ( 1r ‘ 𝑅 )
5 minmar1fval.z ⊢ 0 = ( 0g ‘ 𝑅 )
6 1 2 3 4 5 minmar1val ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
7 6 3expb ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
8 7 3adant3 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
9 simp3l ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → 𝐼 ∈ 𝑁 )
10 simpl3r ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ 𝑖 = 𝐼 ) → 𝐽 ∈ 𝑁 )
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 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
17 eqeq1 ⊢ ( 𝑖 = 𝐼 → ( 𝑖 = 𝐾 ↔ 𝐼 = 𝐾 ) )
18 17 adantr ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → ( 𝑖 = 𝐾 ↔ 𝐼 = 𝐾 ) )
19 eqeq1 ⊢ ( 𝑗 = 𝐽 → ( 𝑗 = 𝐿 ↔ 𝐽 = 𝐿 ) )
20 19 adantl ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → ( 𝑗 = 𝐿 ↔ 𝐽 = 𝐿 ) )
21 20 ifbid ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → if ( 𝑗 = 𝐿 , 1 , 0 ) = if ( 𝐽 = 𝐿 , 1 , 0 ) )
22 oveq12 ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → ( 𝑖 𝑀 𝑗 ) = ( 𝐼 𝑀 𝐽 ) )
23 18 21 22 ifbieq12d ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 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 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 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 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) → ( 𝐼 ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) ) )
26 8 25 mpd ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 1 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )