Metamath Proof Explorer

Theorem minmar1val

Description: Third substitution for the definition 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 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 ) ) ) )

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 minmar1val0 ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
7 6 3ad2ant1 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
8 simp2 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → 𝐾 ∈ 𝑁 )
9 simpl3 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑘 = 𝐾 ) → 𝐿 ∈ 𝑁 )
10 1 2 matrcl ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) )
11 10 simpld ⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin )
12 11 11 jca ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) )
13 12 3ad2ant1 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) )
14 13 adantr ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) )
15 mpoexga ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ V )
16 14 15 syl ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ V )
17 eqeq2 ⊢ ( 𝑘 = 𝐾 → ( 𝑖 = 𝑘 ↔ 𝑖 = 𝐾 ) )
18 17 adantr ⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → ( 𝑖 = 𝑘 ↔ 𝑖 = 𝐾 ) )
19 eqeq2 ⊢ ( 𝑙 = 𝐿 → ( 𝑗 = 𝑙 ↔ 𝑗 = 𝐿 ) )
20 19 ifbid ⊢ ( 𝑙 = 𝐿 → if ( 𝑗 = 𝑙 , 1 , 0 ) = if ( 𝑗 = 𝐿 , 1 , 0 ) )
21 20 adantl ⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → if ( 𝑗 = 𝑙 , 1 , 0 ) = if ( 𝑗 = 𝐿 , 1 , 0 ) )
22 18 21 ifbieq1d ⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) )
23 22 mpoeq3dv ⊢ ( ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
24 23 adantl ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
25 8 9 16 24 ovmpodv2 ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) → ( 𝐾 ( 𝑄 ‘ 𝑀 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
26 7 25 mpd ⊢ ( ( 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 ) ) ) )