Step |
Hyp |
Ref |
Expression |
1 |
|
mendvscafval.a |
⊢ 𝐴 = ( MEndo ‘ 𝑀 ) |
2 |
|
mendvscafval.v |
⊢ · = ( ·𝑠 ‘ 𝑀 ) |
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 |
⊢ ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) |
8 |
1
|
mendbas |
⊢ ( 𝑀 LMHom 𝑀 ) = ( Base ‘ 𝐴 ) |
9 |
3 8
|
eqtr4i |
⊢ 𝐵 = ( 𝑀 LMHom 𝑀 ) |
10 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) |
11 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) |
12 |
|
eqid |
⊢ 𝐵 = 𝐵 |
13 |
6
|
xpeq1i |
⊢ ( 𝐸 × { 𝑥 } ) = ( ( Base ‘ 𝑀 ) × { 𝑥 } ) |
14 |
|
eqid |
⊢ 𝑦 = 𝑦 |
15 |
|
ofeq |
⊢ ( · = ( ·𝑠 ‘ 𝑀 ) → ∘f · = ∘f ( ·𝑠 ‘ 𝑀 ) ) |
16 |
2 15
|
ax-mp |
⊢ ∘f · = ∘f ( ·𝑠 ‘ 𝑀 ) |
17 |
13 14 16
|
oveq123i |
⊢ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) = ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) |
18 |
5 12 17
|
mpoeq123i |
⊢ ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( 𝑥 ∈ ( Base ‘ 𝑆 ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) |
19 |
9 10 11 4 18
|
mendval |
⊢ ( 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) ) |
20 |
19
|
fveq2d |
⊢ ( 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) ) ) |
21 |
5
|
fvexi |
⊢ 𝐾 ∈ V |
22 |
3
|
fvexi |
⊢ 𝐵 ∈ V |
23 |
21 22
|
mpoex |
⊢ ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ∈ V |
24 |
|
eqid |
⊢ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) |
25 |
24
|
algvsca |
⊢ ( ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ∈ V → ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) ) ) |
26 |
23 25
|
mp1i |
⊢ ( 𝑀 ∈ V → ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) 〉 } ) ) ) |
27 |
20 26
|
eqtr4d |
⊢ ( 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ) |
28 |
|
fvprc |
⊢ ( ¬ 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ∅ ) |
29 |
28
|
fveq2d |
⊢ ( ¬ 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( ·𝑠 ‘ ∅ ) ) |
30 |
|
vscaid |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx ) |
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 = ∅ ) → ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ∅ ) |
40 |
38 39
|
syl |
⊢ ( ¬ 𝑀 ∈ V → ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) = ∅ ) |
41 |
32 40
|
eqtr4d |
⊢ ( ¬ 𝑀 ∈ V → ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) ) |
42 |
27 41
|
pm2.61i |
⊢ ( ·𝑠 ‘ ( MEndo ‘ 𝑀 ) ) = ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) |
43 |
7 42
|
eqtri |
⊢ ( ·𝑠 ‘ 𝐴 ) = ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( ( 𝐸 × { 𝑥 } ) ∘f · 𝑦 ) ) |