Theorem ndmovdistr

Description: Any operation is distributive 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
	ndmov.6	⊢ dom 𝐺 = ( 𝑆 × 𝑆 )
Assertion	ndmovdistr	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐹 ( 𝐴 𝐺 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ndmov.1	⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
2		ndmov.5	⊢ ¬ ∅ ∈ 𝑆
3		ndmov.6	⊢ dom 𝐺 = ( 𝑆 × 𝑆 )
4	1 2	ndmovrcl	⊢ ( ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 → ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
5	4	anim2i	⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
6		3anass	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
7	5 6	sylibr	⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
8	3	ndmov	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 𝐹 𝐶 ) ∈ 𝑆 ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ∅ )
9	7 8	nsyl5	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ∅ )
10	3 2	ndmovrcl	⊢ ( ( 𝐴 𝐺 𝐵 ) ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) )
11	3 2	ndmovrcl	⊢ ( ( 𝐴 𝐺 𝐶 ) ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
12	10 11	anim12i	⊢ ( ( ( 𝐴 𝐺 𝐵 ) ∈ 𝑆 ∧ ( 𝐴 𝐺 𝐶 ) ∈ 𝑆 ) → ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
13		anandi3	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ↔ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) )
14	12 13	sylibr	⊢ ( ( ( 𝐴 𝐺 𝐵 ) ∈ 𝑆 ∧ ( 𝐴 𝐺 𝐶 ) ∈ 𝑆 ) → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) )
15	1	ndmov	⊢ ( ¬ ( ( 𝐴 𝐺 𝐵 ) ∈ 𝑆 ∧ ( 𝐴 𝐺 𝐶 ) ∈ 𝑆 ) → ( ( 𝐴 𝐺 𝐵 ) 𝐹 ( 𝐴 𝐺 𝐶 ) ) = ∅ )
16	14 15	nsyl5	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 𝐺 𝐵 ) 𝐹 ( 𝐴 𝐺 𝐶 ) ) = ∅ )
17	9 16	eqtr4d	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐹 ( 𝐴 𝐺 𝐶 ) ) )