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Metamath Proof Explorer

Theorem marepvfval

Description: First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019) (Revised by AV, 26-Feb-2019) (Proof shortened by AV, 2-Mar-2024)

Ref Expression
Hypotheses marepvfval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
marepvfval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
marepvfval.q ⊢ 𝑄 = ( 𝑁 matRepV 𝑅 )
marepvfval.v ⊢ 𝑉 = ( ( Base ‘ 𝑅 ) ↑m 𝑁 )
Assertion marepvfval 𝑄 = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 marepvfval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 marepvfval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
3 marepvfval.q ⊢ 𝑄 = ( 𝑁 matRepV 𝑅 )
4 marepvfval.v ⊢ 𝑉 = ( ( Base ‘ 𝑅 ) ↑m 𝑁 )
5 2 fvexi ⊢ 𝐵 ∈ V
6 4 ovexi ⊢ 𝑉 ∈ V
7 6 a1i ⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝑉 ∈ V )
8 mpoexga ⊢ ( ( 𝐵 ∈ V ∧ 𝑉 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V )
9 5 7 8 sylancr ⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V )
10 oveq12 ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = ( 𝑁 Mat 𝑅 ) )
11 10 1 eqtr4di ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = 𝐴 )
12 11 fveq2d ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = ( Base ‘ 𝐴 ) )
13 12 2 eqtr4di ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = 𝐵 )
14 fveq2 ⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
15 14 adantl ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
16 simpl ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → 𝑛 = 𝑁 )
17 15 16 oveq12d ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = ( ( Base ‘ 𝑅 ) ↑m 𝑁 ) )
18 17 4 eqtr4di ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = 𝑉 )
19 eqidd ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) )
20 16 16 19 mpoeq123dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) )
21 16 20 mpteq12dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑘 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
22 13 18 21 mpoeq123dv ⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑘 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
23 df-marepv ⊢ matRepV = ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑘 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
24 22 23 ovmpoga ⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ∧ ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
25 9 24 mpd3an3 ⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
26 23 mpondm0 ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ∅ )
27 1 fveq2i ⊢ ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
28 2 27 eqtri ⊢ 𝐵 = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
29 matbas0pc ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 Mat 𝑅 ) ) = ∅ )
30 28 29 eqtrid ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
31 30 orcd ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) )
32 0mpo0 ⊢ ( ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ )
33 31 32 syl ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ )
34 26 33 eqtr4d ⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
35 25 34 pm2.61i ⊢ ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
36 3 35 eqtri ⊢ 𝑄 = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )