Step |
Hyp |
Ref |
Expression |
1 |
|
mendplusgfval.a |
⊢ 𝐴 = ( MEndo ‘ 𝑀 ) |
2 |
|
mendplusgfval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
mendplusgfval.p |
⊢ + = ( +g ‘ 𝑀 ) |
4 |
1
|
mendbas |
⊢ ( 𝑀 LMHom 𝑀 ) = ( Base ‘ 𝐴 ) |
5 |
2 4
|
eqtr4i |
⊢ 𝐵 = ( 𝑀 LMHom 𝑀 ) |
6 |
|
ofeq |
⊢ ( + = ( +g ‘ 𝑀 ) → ∘f + = ∘f ( +g ‘ 𝑀 ) ) |
7 |
3 6
|
ax-mp |
⊢ ∘f + = ∘f ( +g ‘ 𝑀 ) |
8 |
7
|
oveqi |
⊢ ( 𝑥 ∘f + 𝑦 ) = ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) |
9 |
8
|
a1i |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∘f + 𝑦 ) = ( 𝑥 ∘f ( +g ‘ 𝑀 ) 𝑦 ) ) |
10 |
9
|
mpoeq3ia |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 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 |
⊢ ( Scalar ‘ 𝑀 ) = ( Scalar ‘ 𝑀 ) |
13 |
|
eqid |
⊢ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) = ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) |
14 |
5 10 11 12 13
|
mendval |
⊢ ( 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) ) |
15 |
1 14
|
syl5eq |
⊢ ( 𝑀 ∈ V → 𝐴 = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) ) |
16 |
15
|
fveq2d |
⊢ ( 𝑀 ∈ V → ( +g ‘ 𝐴 ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) ) ) |
17 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
18 |
17 17
|
mpoex |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) ∈ V |
19 |
|
eqid |
⊢ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) |
20 |
19
|
algaddg |
⊢ ( ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) ) ) |
21 |
18 20
|
mp1i |
⊢ ( 𝑀 ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘ 𝑦 ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑀 ) 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) , 𝑦 ∈ 𝐵 ↦ ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) 〉 } ) ) ) |
22 |
16 21
|
eqtr4d |
⊢ ( 𝑀 ∈ V → ( +g ‘ 𝐴 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) ) |
23 |
|
fvprc |
⊢ ( ¬ 𝑀 ∈ V → ( MEndo ‘ 𝑀 ) = ∅ ) |
24 |
1 23
|
syl5eq |
⊢ ( ¬ 𝑀 ∈ V → 𝐴 = ∅ ) |
25 |
24
|
fveq2d |
⊢ ( ¬ 𝑀 ∈ V → ( +g ‘ 𝐴 ) = ( +g ‘ ∅ ) ) |
26 |
|
df-plusg |
⊢ +g = Slot 2 |
27 |
26
|
str0 |
⊢ ∅ = ( +g ‘ ∅ ) |
28 |
25 27
|
eqtr4di |
⊢ ( ¬ 𝑀 ∈ V → ( +g ‘ 𝐴 ) = ∅ ) |
29 |
24
|
fveq2d |
⊢ ( ¬ 𝑀 ∈ V → ( Base ‘ 𝐴 ) = ( Base ‘ ∅ ) ) |
30 |
|
base0 |
⊢ ∅ = ( Base ‘ ∅ ) |
31 |
29 2 30
|
3eqtr4g |
⊢ ( ¬ 𝑀 ∈ V → 𝐵 = ∅ ) |
32 |
31
|
olcd |
⊢ ( ¬ 𝑀 ∈ V → ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) ) |
33 |
|
0mpo0 |
⊢ ( ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) = ∅ ) |
34 |
32 33
|
syl |
⊢ ( ¬ 𝑀 ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) = ∅ ) |
35 |
28 34
|
eqtr4d |
⊢ ( ¬ 𝑀 ∈ V → ( +g ‘ 𝐴 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) ) |
36 |
22 35
|
pm2.61i |
⊢ ( +g ‘ 𝐴 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 ∘f + 𝑦 ) ) |