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 = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 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 ) → ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V )
9 5 7 8 sylancr ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V )
10 oveq12 ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = ( 𝑁 Mat 𝑅 ) )
11 10 1 eqtr4di ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = 𝐴 )
12 11 fveq2d ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = ( Base ‘ 𝐴 ) )
13 12 2 eqtr4di ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = 𝐵 )
14 fveq2 ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
15 14 adantl ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
16 simpl ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → 𝑛 = 𝑁 )
17 15 16 oveq12d ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = ( ( Base ‘ 𝑅 ) ↑m 𝑁 ) )
18 17 4 eqtr4di ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = 𝑉 )
19 eqidd ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) )
20 16 16 19 mpoeq123dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) )
21 16 20 mpteq12dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑘𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
22 13 18 21 mpoeq123dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑘𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
23 df-marepv matRepV = ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑘𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
24 22 23 ovmpoga ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ∧ ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
25 9 24 mpd3an3 ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 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 syl5eq ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
31 30 orcd ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) )
32 0mpo0 ( ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) → ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ )
33 31 32 syl ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ )
34 26 33 eqtr4d ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
35 25 34 pm2.61i ( 𝑁 matRepV 𝑅 ) = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
36 3 35 eqtri 𝑄 = ( 𝑚𝐵 , 𝑣𝑉 ↦ ( 𝑘𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )