Theorem genpn0

Description: The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-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	genpn0	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ∅ ⊊ ( 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		prn0	⊢ ( 𝐴 ∈ P → 𝐴 ≠ ∅ )
4		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ 𝐴 )
5	3 4	sylib	⊢ ( 𝐴 ∈ P → ∃ 𝑓 𝑓 ∈ 𝐴 )
6		prn0	⊢ ( 𝐵 ∈ P → 𝐵 ≠ ∅ )
7		n0	⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑔 𝑔 ∈ 𝐵 )
8	6 7	sylib	⊢ ( 𝐵 ∈ P → ∃ 𝑔 𝑔 ∈ 𝐵 )
9	5 8	anim12i	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ∃ 𝑓 𝑓 ∈ 𝐴 ∧ ∃ 𝑔 𝑔 ∈ 𝐵 ) )
10	1 2	genpprecl	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵 ) → ( 𝑓 𝐺 𝑔 ) ∈ ( 𝐴 𝐹 𝐵 ) ) )
11		ne0i	⊢ ( ( 𝑓 𝐺 𝑔 ) ∈ ( 𝐴 𝐹 𝐵 ) → ( 𝐴 𝐹 𝐵 ) ≠ ∅ )
12		0pss	⊢ ( ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ↔ ( 𝐴 𝐹 𝐵 ) ≠ ∅ )
13	11 12	sylibr	⊢ ( ( 𝑓 𝐺 𝑔 ) ∈ ( 𝐴 𝐹 𝐵 ) → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) )
14	10 13	syl6	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵 ) → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) )
15	14	expcomd	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝑔 ∈ 𝐵 → ( 𝑓 ∈ 𝐴 → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) ) )
16	15	exlimdv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ∃ 𝑔 𝑔 ∈ 𝐵 → ( 𝑓 ∈ 𝐴 → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) ) )
17	16	com23	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( 𝑓 ∈ 𝐴 → ( ∃ 𝑔 𝑔 ∈ 𝐵 → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) ) )
18	17	exlimdv	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ∃ 𝑓 𝑓 ∈ 𝐴 → ( ∃ 𝑔 𝑔 ∈ 𝐵 → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) ) )
19	18	impd	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ( ( ∃ 𝑓 𝑓 ∈ 𝐴 ∧ ∃ 𝑔 𝑔 ∈ 𝐵 ) → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) ) )
20	9 19	mpd	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) → ∅ ⊊ ( 𝐴 𝐹 𝐵 ) )