Theorem genpss

Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995) (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	genpss	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐴 𝐹 𝐵 ) ⊆ Q )

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	1 2	genpelv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝑓 ∈ ( 𝐴 𝐹 𝐵 ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) ) )
4		elprnq	⊢ ( ( 𝐴 ∈ P ∧ 𝑔 ∈ 𝐴 ) → 𝑔 ∈ Q )
5	4	ex	⊢ ( 𝐴 ∈ P → ( 𝑔 ∈ 𝐴 → 𝑔 ∈ Q ) )
6		elprnq	⊢ ( ( 𝐵 ∈ P ∧ ℎ ∈ 𝐵 ) → ℎ ∈ Q )
7	6	ex	⊢ ( 𝐵 ∈ P → ( ℎ ∈ 𝐵 → ℎ ∈ Q ) )
8	5 7	im2anan9	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ) → ( 𝑔 ∈ Q ∧ ℎ ∈ Q ) ) )
9	2	caovcl	⊢ ( ( 𝑔 ∈ Q ∧ ℎ ∈ Q ) → ( 𝑔 𝐺 ℎ ) ∈ Q )
10	8 9	syl6	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ) → ( 𝑔 𝐺 ℎ ) ∈ Q ) )
11		eleq1a	⊢ ( ( 𝑔 𝐺 ℎ ) ∈ Q → ( 𝑓 = ( 𝑔 𝐺 ℎ ) → 𝑓 ∈ Q ) )
12	10 11	syl6	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ) → ( 𝑓 = ( 𝑔 𝐺 ℎ ) → 𝑓 ∈ Q ) ) )
13	12	rexlimdvv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) → 𝑓 ∈ Q ) )
14	3 13	sylbid	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝑓 ∈ ( 𝐴 𝐹 𝐵 ) → 𝑓 ∈ Q ) )
15	14	ssrdv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐴 𝐹 𝐵 ) ⊆ Q )