Metamath Proof Explorer


Theorem mendvscafval

Description: Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015) (Proof shortened by AV, 3-Mar-2024)

Ref Expression
Hypotheses mendvscafval.a 𝐴 = ( MEndo ‘ 𝑀 )
mendvscafval.v · = ( ·𝑠𝑀 )
mendvscafval.b 𝐵 = ( Base ‘ 𝐴 )
mendvscafval.s 𝑆 = ( Scalar ‘ 𝑀 )
mendvscafval.k 𝐾 = ( Base ‘ 𝑆 )
mendvscafval.e 𝐸 = ( Base ‘ 𝑀 )
Assertion mendvscafval ( ·𝑠𝐴 ) = ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) )

Proof

Step Hyp Ref Expression
1 mendvscafval.a 𝐴 = ( MEndo ‘ 𝑀 )
2 mendvscafval.v · = ( ·𝑠𝑀 )
3 mendvscafval.b 𝐵 = ( Base ‘ 𝐴 )
4 mendvscafval.s 𝑆 = ( Scalar ‘ 𝑀 )
5 mendvscafval.k 𝐾 = ( Base ‘ 𝑆 )
6 mendvscafval.e 𝐸 = ( Base ‘ 𝑀 )
7 1 fveq2i ( ·𝑠𝐴 ) = ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) )
8 1 mendbas ( 𝑀 LMHom 𝑀 ) = ( Base ‘ 𝐴 )
9 3 8 eqtr4i 𝐵 = ( 𝑀 LMHom 𝑀 )
10 eqid ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) )
11 eqid ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) )
12 eqid 𝐵 = 𝐵
13 6 xpeq1i ( 𝐸 × { 𝑥 } ) = ( ( Base ‘ 𝑀 ) × { 𝑥 } )
14 eqid 𝑦 = 𝑦
15 ofeq ( · = ( ·𝑠𝑀 ) → ∘f · = ∘f ( ·𝑠𝑀 ) )
16 2 15 ax-mp f · = ∘f ( ·𝑠𝑀 )
17 13 14 16 oveq123i ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) = ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠𝑀 ) 𝑦 )
18 5 12 17 mpoeq123i ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( 𝑥 ∈ ( Base ‘ 𝑆 ) , 𝑦𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠𝑀 ) 𝑦 ) )
19 9 10 11 4 18 mendval ( 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } ) )
20 19 fveq2d ( 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } ) ) )
21 5 fvexi 𝐾 ∈ V
22 3 fvexi 𝐵 ∈ V
23 21 22 mpoex ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ∈ V
24 eqid ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } )
25 24 algvsca ( ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ∈ V → ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } ) ) )
26 23 25 mp1i ( 𝑀 ∈ V → ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥f ( +g𝑀 ) 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥𝑦 ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ⟩ } ) ) )
27 20 26 eqtr4d ( 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) )
28 fvprc ( ¬ 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ∅ )
29 28 fveq2d ( ¬ 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( ·𝑠 ‘ ∅ ) )
30 df-vsca ·𝑠 = Slot 6
31 30 str0 ∅ = ( ·𝑠 ‘ ∅ )
32 29 31 eqtr4di ( ¬ 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ∅ )
33 fvprc ( ¬ 𝑀 ∈ V → ( Scalar ‘ 𝑀 ) = ∅ )
34 4 33 syl5eq ( ¬ 𝑀 ∈ V → 𝑆 = ∅ )
35 34 fveq2d ( ¬ 𝑀 ∈ V → ( Base ‘ 𝑆 ) = ( Base ‘ ∅ ) )
36 base0 ∅ = ( Base ‘ ∅ )
37 35 5 36 3eqtr4g ( ¬ 𝑀 ∈ V → 𝐾 = ∅ )
38 37 orcd ( ¬ 𝑀 ∈ V → ( 𝐾 = ∅ ∨ 𝐵 = ∅ ) )
39 0mpo0 ( ( 𝐾 = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ∅ )
40 38 39 syl ( ¬ 𝑀 ∈ V → ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ∅ )
41 32 40 eqtr4d ( ¬ 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) )
42 27 41 pm2.61i ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) )
43 7 42 eqtri ( ·𝑠𝐴 ) = ( 𝑥𝐾 , 𝑦𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) )