Theorem genpprecl

Description: Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996) (Revised by Mario Carneiro, 12-Jun-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	genp.1	⊢ 𝐹 = ( 𝑤 ∈ P , 𝑣 ∈ P ↦ { 𝑥 ∣ ∃ 𝑦 ∈ 𝑤 ∃ 𝑧 ∈ 𝑣 𝑥 = ( 𝑦 𝐺 𝑧 ) } )
	genp.2	⊢ ( ( 𝑦 ∈ Q ∧ 𝑧 ∈ Q ) → ( 𝑦 𝐺 𝑧 ) ∈ Q )
Assertion	genpprecl	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 𝐺 𝐷 ) ∈ ( 𝐴 𝐹 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		genp.1	⊢ 𝐹 = ( 𝑤 ∈ P , 𝑣 ∈ P ↦ { 𝑥 ∣ ∃ 𝑦 ∈ 𝑤 ∃ 𝑧 ∈ 𝑣 𝑥 = ( 𝑦 𝐺 𝑧 ) } )
2		genp.2	⊢ ( ( 𝑦 ∈ Q ∧ 𝑧 ∈ Q ) → ( 𝑦 𝐺 𝑧 ) ∈ Q )
3		eqid	⊢ ( 𝐶 𝐺 𝐷 ) = ( 𝐶 𝐺 𝐷 )
4		rspceov	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ ( 𝐶 𝐺 𝐷 ) = ( 𝐶 𝐺 𝐷 ) ) → ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ( 𝐶 𝐺 𝐷 ) = ( 𝑔 𝐺 ℎ ) )
5	3 4	mp3an3	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ( 𝐶 𝐺 𝐷 ) = ( 𝑔 𝐺 ℎ ) )
6	1 2	genpelv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝐶 𝐺 𝐷 ) ∈ ( 𝐴 𝐹 𝐵 ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ( 𝐶 𝐺 𝐷 ) = ( 𝑔 𝐺 ℎ ) ) )
7	5 6	syl5ibr	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 𝐺 𝐷 ) ∈ ( 𝐴 𝐹 𝐵 ) ) )