Step |
Hyp |
Ref |
Expression |
1 |
|
dfgrp3.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
dfgrp3.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
1 2
|
dfgrp3 |
⊢ ( 𝐺 ∈ Grp ↔ ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) |
4 |
|
simp2 |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → 𝐵 ≠ ∅ ) |
5 |
|
sgrpmgm |
⊢ ( 𝐺 ∈ Smgrp → 𝐺 ∈ Mgm ) |
6 |
5
|
adantr |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) → 𝐺 ∈ Mgm ) |
7 |
6
|
adantr |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐺 ∈ Mgm ) |
8 |
|
simpr |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
9 |
8
|
adantr |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
10 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
11 |
1 2
|
mgmcl |
⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
12 |
7 9 10 11
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
13 |
12
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
14 |
1 2
|
sgrpass |
⊢ ( ( 𝐺 ∈ Smgrp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
15 |
14
|
3anassrs |
⊢ ( ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
16 |
15
|
ralrimiva |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
18 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) |
19 |
13 17 18
|
3jca |
⊢ ( ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) |
20 |
19
|
ex |
⊢ ( ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) → ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |
21 |
20
|
ralimdva |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |
22 |
21
|
ralimdva |
⊢ ( 𝐺 ∈ Smgrp → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |
23 |
22
|
a1d |
⊢ ( 𝐺 ∈ Smgrp → ( 𝐵 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) ) |
24 |
23
|
3imp |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) |
25 |
4 24
|
jca |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |
26 |
|
n0 |
⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑎 𝑎 ∈ 𝐵 ) |
27 |
|
3simpa |
⊢ ( ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
28 |
27
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
29 |
1 2
|
issgrpn0 |
⊢ ( 𝑎 ∈ 𝐵 → ( 𝐺 ∈ Smgrp ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) ) |
30 |
28 29
|
syl5ibr |
⊢ ( 𝑎 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → 𝐺 ∈ Smgrp ) ) |
31 |
30
|
exlimiv |
⊢ ( ∃ 𝑎 𝑎 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → 𝐺 ∈ Smgrp ) ) |
32 |
26 31
|
sylbi |
⊢ ( 𝐵 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → 𝐺 ∈ Smgrp ) ) |
33 |
32
|
imp |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) → 𝐺 ∈ Smgrp ) |
34 |
|
simpl |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) → 𝐵 ≠ ∅ ) |
35 |
|
simp3 |
⊢ ( ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) |
36 |
35
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) |
38 |
33 34 37
|
3jca |
⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) → ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) |
39 |
25 38
|
impbii |
⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ↔ ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |
40 |
3 39
|
bitri |
⊢ ( 𝐺 ∈ Grp ↔ ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ∧ ( ∃ 𝑙 ∈ 𝐵 ( 𝑙 + 𝑥 ) = 𝑦 ∧ ∃ 𝑟 ∈ 𝐵 ( 𝑥 + 𝑟 ) = 𝑦 ) ) ) ) |