Step |
Hyp |
Ref |
Expression |
1 |
|
grpinvval.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpinvval.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpinvval.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
grpinvval.n |
⊢ 𝑁 = ( invg ‘ 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 |
8
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 𝑦 + 𝑥 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( 0g ‘ 𝑔 ) = ( 0g ‘ 𝐺 ) ) |
11 |
10 3
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( 0g ‘ 𝑔 ) = 0 ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ↔ ( 𝑦 + 𝑥 ) = 0 ) ) |
13 |
6 12
|
riotaeqbidv |
⊢ ( 𝑔 = 𝐺 → ( ℩ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ) = ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) |
14 |
6 13
|
mpteq12dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
15 |
|
df-minusg |
⊢ invg = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 0g ‘ 𝑔 ) ) ) ) |
16 |
14 15 1
|
mptfvmpt |
⊢ ( 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
17 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ∅ ) |
18 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) = ∅ |
19 |
17 18
|
eqtr4di |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
20 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
21 |
1 20
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ ) |
22 |
21
|
mpteq1d |
⊢ ( ¬ 𝐺 ∈ V → ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) = ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
23 |
19 22
|
eqtr4d |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
24 |
16 23
|
pm2.61i |
⊢ ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) |
25 |
4 24
|
eqtri |
⊢ 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) |