Step |
Hyp |
Ref |
Expression |
1 |
|
grpidval.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpidval.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpidval.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 ) |
6 |
5
|
eleq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑒 ∈ ( Base ‘ 𝑔 ) ↔ 𝑒 ∈ 𝐵 ) ) |
7 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = ( +g ‘ 𝐺 ) ) |
8 |
7 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = + ) |
9 |
8
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = ( 𝑒 + 𝑥 ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ↔ ( 𝑒 + 𝑥 ) = 𝑥 ) ) |
11 |
8
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = ( 𝑥 + 𝑒 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ↔ ( 𝑥 + 𝑒 ) = 𝑥 ) ) |
13 |
10 12
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ↔ ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) |
14 |
5 13
|
raleqbidv |
⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) |
15 |
6 14
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ↔ ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) ) |
16 |
15
|
iotabidv |
⊢ ( 𝑔 = 𝐺 → ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ) = ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) ) |
17 |
|
df-0g |
⊢ 0g = ( 𝑔 ∈ V ↦ ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ) ) |
18 |
|
iotaex |
⊢ ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) ∈ V |
19 |
16 17 18
|
fvmpt |
⊢ ( 𝐺 ∈ V → ( 0g ‘ 𝐺 ) = ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) ) |
20 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( 0g ‘ 𝐺 ) = ∅ ) |
21 |
|
euex |
⊢ ( ∃! 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) → ∃ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) |
22 |
|
n0i |
⊢ ( 𝑒 ∈ 𝐵 → ¬ 𝐵 = ∅ ) |
23 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
24 |
1 23
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ ) |
25 |
22 24
|
nsyl2 |
⊢ ( 𝑒 ∈ 𝐵 → 𝐺 ∈ V ) |
26 |
25
|
adantr |
⊢ ( ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) → 𝐺 ∈ V ) |
27 |
26
|
exlimiv |
⊢ ( ∃ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) → 𝐺 ∈ V ) |
28 |
21 27
|
syl |
⊢ ( ∃! 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) → 𝐺 ∈ V ) |
29 |
|
iotanul |
⊢ ( ¬ ∃! 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) → ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) = ∅ ) |
30 |
28 29
|
nsyl5 |
⊢ ( ¬ 𝐺 ∈ V → ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) = ∅ ) |
31 |
20 30
|
eqtr4d |
⊢ ( ¬ 𝐺 ∈ V → ( 0g ‘ 𝐺 ) = ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) ) |
32 |
19 31
|
pm2.61i |
⊢ ( 0g ‘ 𝐺 ) = ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) |
33 |
3 32
|
eqtri |
⊢ 0 = ( ℩ 𝑒 ( 𝑒 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) ) ) |