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Theorem genpelxp 7466

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 3232	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2		nqex 7318	. . . . . 6 ⊢ Q ∈ V
3	2	elpw2 4141	. . . . 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 145	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
5		ssrab2 3232	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
6	2	elpw2 4141	. . . . 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 145	. . . 4 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ∈ 𝒫 Q
8		opelxpi 4641	. . . 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 424	. . 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 5494	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (1^st ‘𝑤) = (1^st ‘𝐴))
11	10	eleq2d 2240	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (1^st ‘𝑤) ↔ 𝑦 ∈ (1^st ‘𝐴)))
12	11	3anbi1d 1311	. . . . . . 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 2495	. . . . . 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 2719	. . . . 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 5494	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (2^nd ‘𝑤) = (2^nd ‘𝐴))
16	15	eleq2d 2240	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (2^nd ‘𝑤) ↔ 𝑦 ∈ (2^nd ‘𝐴)))
17	16	3anbi1d 1311	. . . . . . 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 2495	. . . . . 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 2719	. . . . 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 3771	. . . 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 5494	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (1^st ‘𝑣) = (1^st ‘𝐵))
22	21	eleq2d 2240	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (1^st ‘𝑣) ↔ 𝑧 ∈ (1^st ‘𝐵)))
23	22	3anbi2d 1312	. . . . . . 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 2495	. . . . . 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 2719	. . . . 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 5494	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (2^nd ‘𝑣) = (2^nd ‘𝐵))
27	26	eleq2d 2240	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (2^nd ‘𝑣) ↔ 𝑧 ∈ (2^nd ‘𝐵)))
28	27	3anbi2d 1312	. . . . . . 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 2495	. . . . . 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 2719	. . . . 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 3771	. . . 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 5985	. . 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 1321	. 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 2261	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 {crab 2452 ⊆ wss 3121 𝒫 cpw 3564 ⟨cop 3584 × cxp 4607 ‘cfv 5196 (class class class)co 5851 ∈ cmpo 5853 1^st c1st 6115 2^nd c2nd 6116 Qcnq 7235 Pcnp 7246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-qs 6517 df-ni 7259 df-nqqs 7303

This theorem is referenced by: addclpr 7492 mulclpr 7527

W3C validator