Theorem genpelv

Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-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	genpelv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐶 ∈ ( 𝐴 𝐹 𝐵 ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) ) )

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	genpv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐴 𝐹 𝐵 ) = { 𝑓 ∣ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) } )
4	3	eleq2d	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐶 ∈ ( 𝐴 𝐹 𝐵 ) ↔ 𝐶 ∈ { 𝑓 ∣ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) } ) )
5		id	⊢ ( 𝐶 = ( 𝑔 𝐺 ℎ ) → 𝐶 = ( 𝑔 𝐺 ℎ ) )
6		ovex	⊢ ( 𝑔 𝐺 ℎ ) ∈ V
7	5 6	eqeltrdi	⊢ ( 𝐶 = ( 𝑔 𝐺 ℎ ) → 𝐶 ∈ V )
8	7	rexlimivw	⊢ ( ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) → 𝐶 ∈ V )
9	8	rexlimivw	⊢ ( ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) → 𝐶 ∈ V )
10		eqeq1	⊢ ( 𝑓 = 𝐶 → ( 𝑓 = ( 𝑔 𝐺 ℎ ) ↔ 𝐶 = ( 𝑔 𝐺 ℎ ) ) )
11	10	2rexbidv	⊢ ( 𝑓 = 𝐶 → ( ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) ) )
12	9 11	elab3	⊢ ( 𝐶 ∈ { 𝑓 ∣ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝑓 = ( 𝑔 𝐺 ℎ ) } ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) )
13	4 12	bitrdi	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝐶 ∈ ( 𝐴 𝐹 𝐵 ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 𝐶 = ( 𝑔 𝐺 ℎ ) ) )