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 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
17 |
|
p0ex |
⊢ { ∅ } ∈ V |
18 |
17 16
|
unex |
⊢ ( { ∅ } ∪ 𝐵 ) ∈ V |
19 |
|
ssun2 |
⊢ 𝐵 ⊆ ( { ∅ } ∪ 𝐵 ) |
20 |
|
riotacl |
⊢ ( ∃! 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ 𝐵 ) |
21 |
19 20
|
sselid |
⊢ ( ∃! 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ ( { ∅ } ∪ 𝐵 ) ) |
22 |
|
ssun1 |
⊢ { ∅ } ⊆ ( { ∅ } ∪ 𝐵 ) |
23 |
|
riotaund |
⊢ ( ¬ ∃! 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) = ∅ ) |
24 |
|
riotaex |
⊢ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ V |
25 |
24
|
elsn |
⊢ ( ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ { ∅ } ↔ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) = ∅ ) |
26 |
23 25
|
sylibr |
⊢ ( ¬ ∃! 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ { ∅ } ) |
27 |
22 26
|
sselid |
⊢ ( ¬ ∃! 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ ( { ∅ } ∪ 𝐵 ) ) |
28 |
21 27
|
pm2.61i |
⊢ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ ( { ∅ } ∪ 𝐵 ) |
29 |
28
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝐵 ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ∈ ( { ∅ } ∪ 𝐵 ) |
30 |
16 18 29
|
mptexw |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ∈ V |
31 |
14 15 30
|
fvmpt |
⊢ ( 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
32 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ∅ ) |
33 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) = ∅ |
34 |
32 33
|
eqtr4di |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
35 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
36 |
1 35
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ ) |
37 |
36
|
mpteq1d |
⊢ ( ¬ 𝐺 ∈ V → ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) = ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
38 |
34 37
|
eqtr4d |
⊢ ( ¬ 𝐺 ∈ V → ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) ) |
39 |
31 38
|
pm2.61i |
⊢ ( invg ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) |
40 |
4 39
|
eqtri |
⊢ 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) ) |