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Theorem genpelxp 7730

Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)

**Hypothesis**
Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)

**Assertion**
Ref	Expression
genpelxp	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Distinct variable groups: 𝑥,𝑦,𝑧,𝑤,𝑣,𝐴 𝑥,𝐵,𝑦,𝑧,𝑤,𝑣 𝑥,𝐺,𝑦,𝑧,𝑤,𝑣 Allowed substitution hints: 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

**Proof of Theorem genpelxp**
Step	Hyp	Ref	Expression
1		ssrab2 3312	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2		nqex 7582	. . . . . 6 ⊢ Q ∈ V
3	2	elpw2 4247	. . . . 5 ⊢ ({𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
4	1, 3	mpbir 146	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
5		ssrab2 3312	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
6	2	elpw2 4247	. . . . 5 ⊢ ({𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ↔ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q)
7	5, 6	mpbir 146	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
8		opelxpi 4757	. . . 4 ⊢ (({𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q ∧ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q) → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q))
9	4, 7, 8	mp2an 426	. . 3 ⊢ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)
10		fveq2 5639	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (1^st ‘𝑤) = (1^st ‘𝐴))
11	10	eleq2d 2301	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (1^st ‘𝑤) ↔ 𝑦 ∈ (1^st ‘𝐴)))
12	11	3anbi1d 1352	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
13	12	2rexbidv 2557	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
14	13	rabbidv 2791	. . . . 5 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
15		fveq2 5639	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (2^nd ‘𝑤) = (2^nd ‘𝐴))
16	15	eleq2d 2301	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (2^nd ‘𝑤) ↔ 𝑦 ∈ (2^nd ‘𝐴)))
17	16	3anbi1d 1352	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
18	17	2rexbidv 2557	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))))
19	18	rabbidv 2791	. . . . 5 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))})
20	14, 19	opeq12d 3870	. . . 4 ⊢ (𝑤 = 𝐴 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
21		fveq2 5639	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (1^st ‘𝑣) = (1^st ‘𝐵))
22	21	eleq2d 2301	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (1^st ‘𝑣) ↔ 𝑧 ∈ (1^st ‘𝐵)))
23	22	3anbi2d 1353	. . . . . . 7 ⊢ (𝑣 = 𝐵 → ((𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
24	23	2rexbidv 2557	. . . . . 6 ⊢ (𝑣 = 𝐵 → (∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
25	24	rabbidv 2791	. . . . 5 ⊢ (𝑣 = 𝐵 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
26		fveq2 5639	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (2^nd ‘𝑣) = (2^nd ‘𝐵))
27	26	eleq2d 2301	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (2^nd ‘𝑣) ↔ 𝑧 ∈ (2^nd ‘𝐵)))
28	27	3anbi2d 1353	. . . . . . 7 ⊢ (𝑣 = 𝐵 → ((𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
29	28	2rexbidv 2557	. . . . . 6 ⊢ (𝑣 = 𝐵 → (∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧)) ↔ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))))
30	29	rabbidv 2791	. . . . 5 ⊢ (𝑣 = 𝐵 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))})
31	25, 30	opeq12d 3870	. . . 4 ⊢ (𝑣 = 𝐵 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
32		genpelvl.1	. . . 4 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
33	20, 31, 32	ovmpog 6155	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩ ∈ (𝒫 Q × 𝒫 Q)) → (𝐴𝐹𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
34	9, 33	mp3an3 1362	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
35	34, 9	eqeltrdi 2322	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 ∃wrex 2511 {crab 2514 ⊆ wss 3200 𝒫 cpw 3652 ⟨cop 3672 × cxp 4723 ‘cfv 5326 (class class class)co 6017 ∈ cmpo 6019 1^st c1st 6300 2^nd c2nd 6301 Qcnq 7499 Pcnp 7510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-qs 6707 df-ni 7523 df-nqqs 7567

This theorem is referenced by: addclpr 7756 mulclpr 7791

W3C validator