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Theorem genpelxp 7452

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 3227	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2		nqex 7304	. . . . . 6 ⊢ Q ∈ V
3	2	elpw2 4136	. . . . 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 3227	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
6	2	elpw2 4136	. . . . 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 4636	. . . 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 423	. . 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 5486	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (1^st ‘𝑤) = (1^st ‘𝐴))
11	10	eleq2d 2236	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (1^st ‘𝑤) ↔ 𝑦 ∈ (1^st ‘𝐴)))
12	11	3anbi1d 1306	. . . . . . 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 2491	. . . . . 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 2715	. . . . 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 5486	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (2^nd ‘𝑤) = (2^nd ‘𝐴))
16	15	eleq2d 2236	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (2^nd ‘𝑤) ↔ 𝑦 ∈ (2^nd ‘𝐴)))
17	16	3anbi1d 1306	. . . . . . 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 2491	. . . . . 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 2715	. . . . 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 3766	. . . 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 5486	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (1^st ‘𝑣) = (1^st ‘𝐵))
22	21	eleq2d 2236	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (1^st ‘𝑣) ↔ 𝑧 ∈ (1^st ‘𝐵)))
23	22	3anbi2d 1307	. . . . . . 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 2491	. . . . . 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 2715	. . . . 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 5486	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (2^nd ‘𝑣) = (2^nd ‘𝐵))
27	26	eleq2d 2236	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (2^nd ‘𝑣) ↔ 𝑧 ∈ (2^nd ‘𝐵)))
28	27	3anbi2d 1307	. . . . . . 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 2491	. . . . . 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 2715	. . . . 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 3766	. . . 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 5976	. . 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 1316	. 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 2257	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∃wrex 2445 {crab 2448 ⊆ wss 3116 𝒫 cpw 3559 ⟨cop 3579 × cxp 4602 ‘cfv 5188 (class class class)co 5842 ∈ cmpo 5844 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 Pcnp 7232

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-qs 6507 df-ni 7245 df-nqqs 7289

This theorem is referenced by: addclpr 7478 mulclpr 7513

W3C validator