Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubval.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubval.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpsubval.i |
⊢ 𝐼 = ( invg ‘ 𝐺 ) |
4 |
|
grpsubval.m |
⊢ − = ( -g ‘ 𝐺 ) |
5 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
6 |
5 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 ) |
7 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = ( +g ‘ 𝐺 ) ) |
8 |
7 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = + ) |
9 |
|
eqidd |
⊢ ( 𝑔 = 𝐺 → 𝑥 = 𝑥 ) |
10 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( invg ‘ 𝑔 ) = ( invg ‘ 𝐺 ) ) |
11 |
10 3
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( invg ‘ 𝑔 ) = 𝐼 ) |
12 |
11
|
fveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( invg ‘ 𝑔 ) ‘ 𝑦 ) = ( 𝐼 ‘ 𝑦 ) ) |
13 |
8 9 12
|
oveq123d |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( +g ‘ 𝑔 ) ( ( invg ‘ 𝑔 ) ‘ 𝑦 ) ) = ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) |
14 |
6 6 13
|
mpoeq123dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( +g ‘ 𝑔 ) ( ( invg ‘ 𝑔 ) ‘ 𝑦 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) ) |
15 |
|
df-sbg |
⊢ -g = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( +g ‘ 𝑔 ) ( ( invg ‘ 𝑔 ) ‘ 𝑦 ) ) ) ) |
16 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
17 |
2
|
fvexi |
⊢ + ∈ V |
18 |
17
|
rnex |
⊢ ran + ∈ V |
19 |
|
p0ex |
⊢ { ∅ } ∈ V |
20 |
18 19
|
unex |
⊢ ( ran + ∪ { ∅ } ) ∈ V |
21 |
|
df-ov |
⊢ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) = ( + ‘ 〈 𝑥 , ( 𝐼 ‘ 𝑦 ) 〉 ) |
22 |
|
fvrn0 |
⊢ ( + ‘ 〈 𝑥 , ( 𝐼 ‘ 𝑦 ) 〉 ) ∈ ( ran + ∪ { ∅ } ) |
23 |
21 22
|
eqeltri |
⊢ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ∈ ( ran + ∪ { ∅ } ) |
24 |
23
|
rgen2w |
⊢ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ∈ ( ran + ∪ { ∅ } ) |
25 |
16 16 20 24
|
mpoexw |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) ∈ V |
26 |
14 15 25
|
fvmpt |
⊢ ( 𝐺 ∈ V → ( -g ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) ) |
27 |
4 26
|
eqtrid |
⊢ ( 𝐺 ∈ V → − = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) ) |
28 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( -g ‘ 𝐺 ) = ∅ ) |
29 |
4 28
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → − = ∅ ) |
30 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
31 |
1 30
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ ) |
32 |
31
|
olcd |
⊢ ( ¬ 𝐺 ∈ V → ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) ) |
33 |
|
0mpo0 |
⊢ ( ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) = ∅ ) |
34 |
32 33
|
syl |
⊢ ( ¬ 𝐺 ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) = ∅ ) |
35 |
29 34
|
eqtr4d |
⊢ ( ¬ 𝐺 ∈ V → − = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) ) |
36 |
27 35
|
pm2.61i |
⊢ − = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + ( 𝐼 ‘ 𝑦 ) ) ) |