Step |
Hyp |
Ref |
Expression |
1 |
|
ndmov.1 |
⊢ dom 𝐹 = ( 𝑆 × 𝑆 ) |
2 |
|
ndmov.5 |
⊢ ¬ ∅ ∈ 𝑆 |
3 |
1 2
|
ndmovrcl |
⊢ ( ( 𝐴 𝐹 𝐵 ) ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) |
4 |
3
|
anim1i |
⊢ ( ( ( 𝐴 𝐹 𝐵 ) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝐶 ∈ 𝑆 ) ) |
5 |
|
df-3an |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝐶 ∈ 𝑆 ) ) |
6 |
4 5
|
sylibr |
⊢ ( ( ( 𝐴 𝐹 𝐵 ) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) |
7 |
1
|
ndmov |
⊢ ( ¬ ( ( 𝐴 𝐹 𝐵 ) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ∅ ) |
8 |
6 7
|
nsyl5 |
⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ∅ ) |
9 |
1 2
|
ndmovrcl |
⊢ ( ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 → ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) |
10 |
9
|
anim2i |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) ) |
11 |
|
3anass |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) ) |
12 |
10 11
|
sylibr |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) |
13 |
1
|
ndmov |
⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) = ∅ ) |
14 |
12 13
|
nsyl5 |
⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) = ∅ ) |
15 |
8 14
|
eqtr4d |
⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) ) |