Step |
Hyp |
Ref |
Expression |
1 |
|
mgm2nsgrp.s |
⊢ 𝑆 = { 𝐴 , 𝐵 } |
2 |
|
mgm2nsgrp.b |
⊢ ( Base ‘ 𝑀 ) = 𝑆 |
3 |
|
mgm2nsgrp.o |
⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝑆 , 𝑦 ∈ 𝑆 ↦ if ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ) , 𝐵 , 𝐴 ) ) |
4 |
|
prid1g |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 , 𝐵 } ) |
5 |
4 1
|
eleqtrrdi |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆 ) |
6 |
|
prid2g |
⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ { 𝐴 , 𝐵 } ) |
7 |
6 1
|
eleqtrrdi |
⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆 ) |
8 |
2
|
eqcomi |
⊢ 𝑆 = ( Base ‘ 𝑀 ) |
9 |
|
ne0i |
⊢ ( 𝐴 ∈ 𝑆 → 𝑆 ≠ ∅ ) |
10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → 𝑆 ≠ ∅ ) |
11 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝐵 ∈ 𝑆 ) |
12 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝐴 ∈ 𝑆 ) |
13 |
8 3 10 11 12
|
opifismgm |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → 𝑀 ∈ Mgm ) |
14 |
5 7 13
|
syl2an |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝑀 ∈ Mgm ) |