Metamath Proof Explorer

Theorem ndmaovass

Description: Any operation is associative outside its domain. In contrast to ndmovass where it is required that the operation's domain doesn't contain the empty set ( -. (/) e. S ), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017)

Ref Expression
Hypothesis ndmaov.1 ⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
Assertion ndmaovass ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → (( (( 𝐴 𝐹 𝐵 )) 𝐹 𝐶 )) = (( 𝐴 𝐹 (( 𝐵 𝐹 𝐶 )) )) )

Proof

Step Hyp Ref Expression
1 ndmaov.1 ⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
2 1 eleq2i ⊢ ( ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ dom 𝐹 ↔ ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ ( 𝑆 × 𝑆 ) )
3 opelxp ⊢ ( ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ ( 𝑆 × 𝑆 ) ↔ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
4 2 3 bitri ⊢ ( ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ dom 𝐹 ↔ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
5 aovvdm ⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝑆 → ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 )
6 1 eleq2i ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑆 × 𝑆 ) )
7 opelxp ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑆 × 𝑆 ) ↔ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) )
8 6 7 bitri ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ↔ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) )
9 df-3an ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝐶 ∈ 𝑆 ) )
10 9 simplbi2 ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐶 ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
11 8 10 sylbi ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 → ( 𝐶 ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
12 5 11 syl ⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝑆 → ( 𝐶 ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
13 12 imp ⊢ ( ( (( 𝐴 𝐹 𝐵 )) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
14 4 13 sylbi ⊢ ( ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
15 ndmaov ⊢ ( ¬ ⟨ (( 𝐴 𝐹 𝐵 )) , 𝐶 ⟩ ∈ dom 𝐹 → (( (( 𝐴 𝐹 𝐵 )) 𝐹 𝐶 )) = V )
16 14 15 nsyl5 ⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → (( (( 𝐴 𝐹 𝐵 )) 𝐹 𝐶 )) = V )
17 1 eleq2i ⊢ ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 ↔ ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ ( 𝑆 × 𝑆 ) )
18 opelxp ⊢ ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ ( 𝑆 × 𝑆 ) ↔ ( 𝐴 ∈ 𝑆 ∧ (( 𝐵 𝐹 𝐶 )) ∈ 𝑆 ) )
19 17 18 bitri ⊢ ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 ↔ ( 𝐴 ∈ 𝑆 ∧ (( 𝐵 𝐹 𝐶 )) ∈ 𝑆 ) )
20 aovvdm ⊢ ( (( 𝐵 𝐹 𝐶 )) ∈ 𝑆 → ⟨ 𝐵 , 𝐶 ⟩ ∈ dom 𝐹 )
21 1 eleq2i ⊢ ( ⟨ 𝐵 , 𝐶 ⟩ ∈ dom 𝐹 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ ( 𝑆 × 𝑆 ) )
22 opelxp ⊢ ( ⟨ 𝐵 , 𝐶 ⟩ ∈ ( 𝑆 × 𝑆 ) ↔ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
23 21 22 bitri ⊢ ( ⟨ 𝐵 , 𝐶 ⟩ ∈ dom 𝐹 ↔ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
24 3anass ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
25 24 biimpri ⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
26 25 a1d ⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
27 26 expcom ⊢ ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) ) )
28 23 27 sylbi ⊢ ( ⟨ 𝐵 , 𝐶 ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) ) )
29 20 28 syl ⊢ ( (( 𝐵 𝐹 𝐶 )) ∈ 𝑆 → ( 𝐴 ∈ 𝑆 → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) ) )
30 29 impcom ⊢ ( ( 𝐴 ∈ 𝑆 ∧ (( 𝐵 𝐹 𝐶 )) ∈ 𝑆 ) → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
31 19 30 sylbi ⊢ ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
32 31 pm2.43i ⊢ ( ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
33 ndmaov ⊢ ( ¬ ⟨ 𝐴 , (( 𝐵 𝐹 𝐶 )) ⟩ ∈ dom 𝐹 → (( 𝐴 𝐹 (( 𝐵 𝐹 𝐶 )) )) = V )
34 32 33 nsyl5 ⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → (( 𝐴 𝐹 (( 𝐵 𝐹 𝐶 )) )) = V )
35 16 34 eqtr4d ⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → (( (( 𝐴 𝐹 𝐵 )) 𝐹 𝐶 )) = (( 𝐴 𝐹 (( 𝐵 𝐹 𝐶 )) )) )