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 |
1
|
hashprdifel |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ) |
5 |
|
simp1 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ 𝑆 ) |
6 |
|
simp2 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ∈ 𝑆 ) |
7 |
5 5 6
|
3jca |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) |
8 |
4 7
|
syl |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) |
9 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ≠ 𝐵 ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
11 |
1 2 3 10
|
mgm2nsgrplem2 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ( +g ‘ 𝑀 ) 𝐵 ) = 𝐴 ) |
12 |
11
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ( +g ‘ 𝑀 ) 𝐵 ) = 𝐴 ) |
13 |
1 2 3 10
|
mgm2nsgrplem3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ( +g ‘ 𝑀 ) ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) = 𝐵 ) |
14 |
13
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 ( +g ‘ 𝑀 ) ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) = 𝐵 ) |
15 |
9 12 14
|
3netr4d |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ( +g ‘ 𝑀 ) 𝐵 ) ≠ ( 𝐴 ( +g ‘ 𝑀 ) ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) ) |
16 |
4 15
|
syl |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ( +g ‘ 𝑀 ) 𝐵 ) ≠ ( 𝐴 ( +g ‘ 𝑀 ) ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) ) |
17 |
2
|
eqcomi |
⊢ 𝑆 = ( Base ‘ 𝑀 ) |
18 |
17 10
|
isnsgrp |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ( +g ‘ 𝑀 ) 𝐵 ) ≠ ( 𝐴 ( +g ‘ 𝑀 ) ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) → 𝑀 ∉ Smgrp ) ) |
19 |
8 16 18
|
sylc |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → 𝑀 ∉ Smgrp ) |