Theorem genipv 7122

Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)

Hypotheses

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

Assertion

Ref	Expression
genipv	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑞,𝑟,𝑠,𝐴   𝑥,𝐵,𝑦,𝑧,𝑞,𝑟,𝑠   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑞,𝑟,𝑠

Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑠,𝑟,𝑞)

Proof of Theorem genipv

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5673	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2		fveq2 5318	. . . . . . 7 ⊢ (𝑓 = 𝐴 → (1st ‘𝑓) = (1st ‘𝐴))
3	2	rexeqdv 2570	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
4	3	rabbidv 2609	. . . . 5 ⊢ (𝑓 = 𝐴 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)})
5		fveq2 5318	. . . . . . 7 ⊢ (𝑓 = 𝐴 → (2nd ‘𝑓) = (2nd ‘𝐴))
6	5	rexeqdv 2570	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
7	6	rabbidv 2609	. . . . 5 ⊢ (𝑓 = 𝐴 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)})
8	4, 7	opeq12d 3636	. . . 4 ⊢ (𝑓 = 𝐴 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
9	1, 8	eqeq12d 2103	. . 3 ⊢ (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ↔ (𝐴𝐹𝑔) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩))
10		oveq2 5674	. . . 4 ⊢ (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
11		fveq2 5318	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (1st ‘𝑔) = (1st ‘𝐵))
12	11	rexeqdv 2570	. . . . . . 7 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)))
13	12	rexbidv 2382	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)))
14	13	rabbidv 2609	. . . . 5 ⊢ (𝑔 = 𝐵 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)})
15		fveq2 5318	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (2nd ‘𝑔) = (2nd ‘𝐵))
16	15	rexeqdv 2570	. . . . . . 7 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)))
17	16	rexbidv 2382	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)))
18	17	rabbidv 2609	. . . . 5 ⊢ (𝑔 = 𝐵 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)})
19	14, 18	opeq12d 3636	. . . 4 ⊢ (𝑔 = 𝐵 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
20	10, 19	eqeq12d 2103	. . 3 ⊢ (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ↔ (𝐴𝐹𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)}⟩))
21		nqex 6976	. . . . . . 7 ⊢ Q ∈ V
22	21	a1i 9	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → Q ∈ V)
23		rabssab 3109	. . . . . . 7 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}
24		prop 7088	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ P → ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩ ∈ P)
25		elprnql 7094	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑓), (2nd ‘𝑓)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝑓)) → 𝑦 ∈ Q)
26	24, 25	sylan 278	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ (1st ‘𝑓)) → 𝑦 ∈ Q)
27		prop 7088	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ P → ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∈ P)
28		elprnql 7094	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝑔)) → 𝑧 ∈ Q)
29	27, 28	sylan 278	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ (1st ‘𝑔)) → 𝑧 ∈ Q)
30		genp.2	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
31		eleq1 2151	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦𝐺𝑧) → (𝑥 ∈ Q ↔ (𝑦𝐺𝑧) ∈ Q))
32	30, 31	syl5ibrcom 156	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
33	26, 29, 32	syl2an 284	. . . . . . . . . 10 ⊢ (((𝑓 ∈ P ∧ 𝑦 ∈ (1st ‘𝑓)) ∧ (𝑔 ∈ P ∧ 𝑧 ∈ (1st ‘𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
34	33	an4s 556	. . . . . . . . 9 ⊢ (((𝑓 ∈ P ∧ 𝑔 ∈ P) ∧ (𝑦 ∈ (1st ‘𝑓) ∧ 𝑧 ∈ (1st ‘𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
35	34	rexlimdvva 2497	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
36	35	abssdv 3096	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
37	23, 36	syl5ss 3037	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
38	22, 37	ssexd 3985	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
39		rabssab 3109	. . . . . . 7 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}
40		elprnqu 7095	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑓), (2nd ‘𝑓)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝑓)) → 𝑦 ∈ Q)
41	24, 40	sylan 278	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ (2nd ‘𝑓)) → 𝑦 ∈ Q)
42		elprnqu 7095	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝑔)) → 𝑧 ∈ Q)
43	27, 42	sylan 278	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ (2nd ‘𝑔)) → 𝑧 ∈ Q)
44	41, 43, 32	syl2an 284	. . . . . . . . . 10 ⊢ (((𝑓 ∈ P ∧ 𝑦 ∈ (2nd ‘𝑓)) ∧ (𝑔 ∈ P ∧ 𝑧 ∈ (2nd ‘𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
45	44	an4s 556	. . . . . . . . 9 ⊢ (((𝑓 ∈ P ∧ 𝑔 ∈ P) ∧ (𝑦 ∈ (2nd ‘𝑓) ∧ 𝑧 ∈ (2nd ‘𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
46	45	rexlimdvva 2497	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
47	46	abssdv 3096	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
48	39, 47	syl5ss 3037	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
49	22, 48	ssexd 3985	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
50		opelxp 4480	. . . . 5 ⊢ (⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V) ↔ ({𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V ∧ {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V))
51	38, 49, 50	sylanbrc 409	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V))
52		fveq2 5318	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → (1st ‘𝑤) = (1st ‘𝑓))
53	52	rexeqdv 2570	. . . . . . 7 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ (1st ‘𝑤)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)))
54	53	rabbidv 2609	. . . . . 6 ⊢ (𝑤 = 𝑓 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑤)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)})
55		fveq2 5318	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → (2nd ‘𝑤) = (2nd ‘𝑓))
56	55	rexeqdv 2570	. . . . . . 7 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ (2nd ‘𝑤)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)))
57	56	rabbidv 2609	. . . . . 6 ⊢ (𝑤 = 𝑓 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑤)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)})
58	54, 57	opeq12d 3636	. . . . 5 ⊢ (𝑤 = 𝑓 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑤)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑤)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
59		fveq2 5318	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (1st ‘𝑣) = (1st ‘𝑔))
60	59	rexeqdv 2570	. . . . . . . 8 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
61	60	rexbidv 2382	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
62	61	rabbidv 2609	. . . . . 6 ⊢ (𝑣 = 𝑔 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)})
63		fveq2 5318	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (2nd ‘𝑣) = (2nd ‘𝑔))
64	63	rexeqdv 2570	. . . . . . . 8 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
65	64	rexbidv 2382	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)))
66	65	rabbidv 2609	. . . . . 6 ⊢ (𝑣 = 𝑔 → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)})
67	62, 66	opeq12d 3636	. . . . 5 ⊢ (𝑣 = 𝑔 → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
68		genp.1	. . . . . 6 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
69	68	genpdf 7121	. . . . 5 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑤)∃𝑧 ∈ (1st ‘𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑤)∃𝑧 ∈ (2nd ‘𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
70	58, 67, 69	ovmpt2g 5793	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V)) → (𝑓𝐹𝑔) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
71	51, 70	mpd3an3 1275	. . 3 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝑓)∃𝑧 ∈ (1st ‘𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝑓)∃𝑧 ∈ (2nd ‘𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
72	9, 20, 71	vtocl2ga 2688	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
73		eqeq1 2095	. . . . . 6 ⊢ (𝑥 = 𝑞 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑦𝐺𝑧)))
74	73	2rexbidv 2404	. . . . 5 ⊢ (𝑥 = 𝑞 → (∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑞 = (𝑦𝐺𝑧)))
75		oveq1 5673	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑦𝐺𝑧) = (𝑟𝐺𝑧))
76	75	eqeq2d 2100	. . . . . 6 ⊢ (𝑦 = 𝑟 → (𝑞 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑧)))
77		oveq2 5674	. . . . . . 7 ⊢ (𝑧 = 𝑠 → (𝑟𝐺𝑧) = (𝑟𝐺𝑠))
78	77	eqeq2d 2100	. . . . . 6 ⊢ (𝑧 = 𝑠 → (𝑞 = (𝑟𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑠)))
79	76, 78	cbvrex2v 2600	. . . . 5 ⊢ (∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠))
80	74, 79	syl6bb 195	. . . 4 ⊢ (𝑥 = 𝑞 → (∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)))
81	80	cbvrabv 2619	. . 3 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}
82	73	2rexbidv 2404	. . . . 5 ⊢ (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑞 = (𝑦𝐺𝑧)))
83	76, 78	cbvrex2v 2600	. . . . 5 ⊢ (∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠))
84	82, 83	syl6bb 195	. . . 4 ⊢ (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)))
85	84	cbvrabv 2619	. . 3 ⊢ {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}
86	81, 85	opeq12i 3633	. 2 ⊢ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}⟩
87	72, 86	syl6eq 2137	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  {cab 2075  ∃wrex 2361  {crab 2364  Vcvv 2620  ⟨cop 3453   × cxp 4449  ‘cfv 5028  (class class class)co 5666   ↦ cmpt2 5668  1^st c1st 5923  2^nd c2nd 5924  Qcnq 6893  Pcnp 6904

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6917  df-nqqs 6961  df-inp 7079

This theorem is referenced by:  genpelvl  7125  genpelvu  7126  plpvlu  7151  mpvlu  7152