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Metamath Proof Explorer

Theorem minmar1fval

Description: First 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 minmar1fval 𝑄 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 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 oveq12 ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = ( 𝑁 Mat 𝑅 ) )
7 6 1 eqtr4di ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = 𝐴 )
8 7 fveq2d ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = ( Base ‘ 𝐴 ) )
9 8 2 eqtr4di ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = 𝐵 )
10 simpl ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → 𝑛 = 𝑁 )
11 fveq2 ⊢ ( 𝑟 = 𝑅 → ( 1r ‘ 𝑟 ) = ( 1r ‘ 𝑅 ) )
12 11 4 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( 1r ‘ 𝑟 ) = 1 )
13 fveq2 ⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = ( 0g ‘ 𝑅 ) )
14 13 5 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = 0 )
15 12 14 ifeq12d ⊢ ( 𝑟 = 𝑅 → if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) = if ( 𝑗 = 𝑙 , 1 , 0 ) )
16 15 ifeq1d ⊢ ( 𝑟 = 𝑅 → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) )
17 16 adantl ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) )
18 10 10 17 mpoeq123dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) )
19 10 10 18 mpoeq123dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑘 ∈ 𝑛 , 𝑙 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
20 9 19 mpteq12dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑘 ∈ 𝑛 , 𝑙 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
21 df-minmar1 ⊢ minMatR1 = ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑘 ∈ 𝑛 , 𝑙 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
22 2 fvexi ⊢ 𝐵 ∈ V
23 22 mptex ⊢ ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V
24 20 21 23 ovmpoa ⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
25 21 mpondm0 ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ∅ )
26 mpt0 ⊢ ( 𝑚 ∈ ∅ ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅
27 25 26 eqtr4di ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚 ∈ ∅ ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
28 1 fveq2i ⊢ ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
29 2 28 eqtri ⊢ 𝐵 = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
30 matbas0pc ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 Mat 𝑅 ) ) = ∅ )
31 29 30 eqtrid ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
32 31 mpteq1d ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚 ∈ ∅ ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
33 27 32 eqtr4d ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
34 24 33 pm2.61i ⊢ ( 𝑁 minMatR1 𝑅 ) = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
35 3 34 eqtri ⊢ 𝑄 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )