Step |
Hyp |
Ref |
Expression |
1 |
|
grppropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
grppropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
grppropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
1 2 3
|
mndpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd ) ) |
5 |
1 2 3
|
grpidpropd |
⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) ) |
7 |
3 6
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
8 |
7
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
9 |
8
|
rexbidva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
10 |
9
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
11 |
1
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) ) |
12 |
1 11
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) ) |
13 |
2
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
14 |
2 13
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
15 |
10 12 14
|
3bitr3d |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
16 |
4 15
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
18 |
|
eqid |
⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐾 ) |
19 |
|
eqid |
⊢ ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐾 ) |
20 |
17 18 19
|
isgrp |
⊢ ( 𝐾 ∈ Grp ↔ ( 𝐾 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
22 |
|
eqid |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 ) |
24 |
21 22 23
|
isgrp |
⊢ ( 𝐿 ∈ Grp ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) |
25 |
16 20 24
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) ) |