Theorem ndmovass

Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995)

	Ref	Expression
Hypotheses	ndmov.1	⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
	ndmov.5	⊢ ¬ ∅ ∈ 𝑆
Assertion	ndmovass	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )

Proof

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 ) ) )