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Theorem genpelxp 7319

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 3182	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
2		nqex 7171	. . . . . 6 ⊢ Q ∈ V
3	2	elpw2 4082	. . . . 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 3182	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (2^nd ‘𝐵) ∧ 𝑥 = (𝑦𝐺𝑧))} ⊆ Q
6	2	elpw2 4082	. . . . 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 4571	. . . 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 422	. . 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 5421	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (1^st ‘𝑤) = (1^st ‘𝐴))
11	10	eleq2d 2209	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (1^st ‘𝑤) ↔ 𝑦 ∈ (1^st ‘𝐴)))
12	11	3anbi1d 1294	. . . . . . 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 2460	. . . . . 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 2675	. . . . 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 5421	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (2^nd ‘𝑤) = (2^nd ‘𝐴))
16	15	eleq2d 2209	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ (2^nd ‘𝑤) ↔ 𝑦 ∈ (2^nd ‘𝐴)))
17	16	3anbi1d 1294	. . . . . . 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 2460	. . . . . 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 2675	. . . . 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 3713	. . . 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 5421	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (1^st ‘𝑣) = (1^st ‘𝐵))
22	21	eleq2d 2209	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (1^st ‘𝑣) ↔ 𝑧 ∈ (1^st ‘𝐵)))
23	22	3anbi2d 1295	. . . . . . 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 2460	. . . . . 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 2675	. . . . 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 5421	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (2^nd ‘𝑣) = (2^nd ‘𝐵))
27	26	eleq2d 2209	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝑧 ∈ (2^nd ‘𝑣) ↔ 𝑧 ∈ (2^nd ‘𝐵)))
28	27	3anbi2d 1295	. . . . . . 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 2460	. . . . . 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 2675	. . . . 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 3713	. . . 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 5905	. . 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 1304	. 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 2230	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 {crab 2420 ⊆ wss 3071 𝒫 cpw 3510 ⟨cop 3530 × cxp 4537 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 1^st c1st 6036 2^nd c2nd 6037 Qcnq 7088 Pcnp 7099

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-qs 6435 df-ni 7112 df-nqqs 7156

This theorem is referenced by: addclpr 7345 mulclpr 7380

W3C validator