Step |
Hyp |
Ref |
Expression |
1 |
|
mgm2nsgrp.s |
⊢ 𝑆 = { 𝐴 , 𝐵 } |
2 |
|
mgm2nsgrp.b |
⊢ ( Base ‘ 𝑀 ) = 𝑆 |
3 |
|
sgrp2nmnd.o |
⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝑆 , 𝑦 ∈ 𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) ) |
4 |
|
sgrp2nmnd.p |
⊢ ⚬ = ( +g ‘ 𝑀 ) |
5 |
4 3
|
eqtri |
⊢ ⚬ = ( 𝑥 ∈ 𝑆 , 𝑦 ∈ 𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) ) |
6 |
5
|
a1i |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ⚬ = ( 𝑥 ∈ 𝑆 , 𝑦 ∈ 𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) ) ) |
7 |
|
df-ne |
⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 ) |
8 |
|
eqeq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐴 = 𝑥 ↔ 𝐴 = 𝐵 ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ( 𝐴 = 𝑥 ↔ 𝐴 = 𝐵 ) ) |
10 |
|
eqcom |
⊢ ( 𝐴 = 𝑥 ↔ 𝑥 = 𝐴 ) |
11 |
9 10
|
bitr3di |
⊢ ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ( 𝐴 = 𝐵 ↔ 𝑥 = 𝐴 ) ) |
12 |
11
|
notbid |
⊢ ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ( ¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴 ) ) |
13 |
12
|
biimpcd |
⊢ ( ¬ 𝐴 = 𝐵 → ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) ) |
14 |
7 13
|
sylbi |
⊢ ( 𝐴 ≠ 𝐵 → ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) ) |
15 |
14
|
3ad2ant3 |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) ) |
16 |
15
|
imp |
⊢ ( ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) ) → ¬ 𝑥 = 𝐴 ) |
17 |
16
|
iffalsed |
⊢ ( ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ) ) → if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) = 𝐵 ) |
18 |
|
simp2 |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ∈ 𝑆 ) |
19 |
|
simp1 |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐶 ∈ 𝑆 ) |
20 |
6 17 18 19 18
|
ovmpod |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ⚬ 𝐶 ) = 𝐵 ) |