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Metamath Proof Explorer

Theorem maducoeval

Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018)

Ref Expression
Hypotheses madufval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
madufval.d ⊢ 𝐷 = ( 𝑁 maDet 𝑅 )
madufval.j ⊢ 𝐽 = ( 𝑁 maAdju 𝑅 )
madufval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
madufval.o ⊢ 1 = ( 1r ‘ 𝑅 )
madufval.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion maducoeval ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 madufval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 madufval.d ⊢ 𝐷 = ( 𝑁 maDet 𝑅 )
3 madufval.j ⊢ 𝐽 = ( 𝑁 maAdju 𝑅 )
4 madufval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
5 madufval.o ⊢ 1 = ( 1r ‘ 𝑅 )
6 madufval.z ⊢ 0 = ( 0g ‘ 𝑅 )
7 1 2 3 4 5 6 maduval ⊢ ( 𝑀 ∈ 𝐵 → ( 𝐽 ‘ 𝑀 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) )
8 7 3ad2ant1 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐽 ‘ 𝑀 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) )
9 simp1r ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → 𝑗 = 𝐻 )
10 9 eqeq2d ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑘 = 𝑗 ↔ 𝑘 = 𝐻 ) )
11 simp1l ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → 𝑖 = 𝐼 )
12 11 eqeq2d ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑙 = 𝑖 ↔ 𝑙 = 𝐼 ) )
13 12 ifbid ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝑖 , 1 , 0 ) = if ( 𝑙 = 𝐼 , 1 , 0 ) )
14 10 13 ifbieq1d ⊢ ( ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) )
15 14 mpoeq3dva ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) )
16 15 fveq2d ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) )
17 16 adantl ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐻 ) ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑗 , if ( 𝑙 = 𝑖 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) )
18 simp2 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → 𝐼 ∈ 𝑁 )
19 simp3 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → 𝐻 ∈ 𝑁 )
20 fvexd ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ∈ V )
21 8 17 18 19 20 ovmpod ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) )