Theorem genprndu 7178

Description: The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)

Hypotheses

Ref	Expression
genpelvl.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))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genprndu.ord	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
genprndu.com	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
genprndu.upper	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))

Assertion

Ref	Expression
genprndu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞   𝑥,𝐺,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝑞   𝑔,𝐹,𝑞   𝐴,𝑟,𝑞,𝑣,𝑤,𝑥,𝑦,𝑧   𝐵,𝑟,𝑔,ℎ   ℎ,𝐹,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧   𝐺,𝑟

Proof of Theorem genprndu

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . . . . 10 ⊢ 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))}⟩)
2		genpelvl.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelvu 7169	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd ‘𝐴)∃𝑏 ∈ (2nd ‘𝐵)𝑟 = (𝑎𝐺𝑏)))
4		r2ex 2409	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (2nd ‘𝐴)∃𝑏 ∈ (2nd ‘𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
5	3, 4	syl6bb 195	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
6	5	biimpa 291	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
7	6	adantrl 463	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8		prop 7131	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
9		prnminu 7145	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (2nd ‘𝐴)) → ∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎)
10	8, 9	sylan 278	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝑎 ∈ (2nd ‘𝐴)) → ∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎)
11		prop 7131	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12		prnminu 7145	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (2nd ‘𝐵)) → ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏)
13	11, 12	sylan 278	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑏 ∈ (2nd ‘𝐵)) → ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏)
14	10, 13	anim12i 332	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝑎 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑏 ∈ (2nd ‘𝐵))) → (∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏))
15	14	an4s 556	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵))) → (∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏))
16		reeanv 2550	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) ↔ (∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏))
17	15, 16	sylibr 133	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏))
18		genprndu.ord	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
19		genprndu.com	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
20	18, 19	genplt2i 7166	. . . . . . . . . . . . . 14 ⊢ ((𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
21	20	reximi 2482	. . . . . . . . . . . . 13 ⊢ (∃𝑑 ∈ (2nd ‘𝐵)(𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
22	21	reximi 2482	. . . . . . . . . . . 12 ⊢ (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
23	17, 22	syl 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
24	23	adantrr 464	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25		breq2 3871	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
26	25	biimprd 157	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
27	26	reximdv 2486	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
28	27	reximdv 2486	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
29	28	ad2antll 476	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
30	24, 29	mpd 13	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟)
31	30	ex 114	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
32	31	exlimdvv 1832	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
33	32	adantr 271	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
34	7, 33	mpd 13	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟)
35	1, 2	genppreclu 7171	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
36	35	imp 123	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37		elprnqu 7138	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑐 ∈ Q)
38	8, 37	sylan 278	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑐 ∈ Q)
39		elprnqu 7138	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑑 ∈ Q)
40	11, 39	sylan 278	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑑 ∈ Q)
41	38, 40	anim12i 332	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
42	41	an4s 556	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
43	2	caovcl 5837	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Q ∧ 𝑑 ∈ Q) → (𝑐𝐺𝑑) ∈ Q)
44	42, 43	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45		breq1 3870	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46		eleq1 2157	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
47	45, 46	anbi12d 458	. . . . . . . . . 10 ⊢ (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
48	47	adantl 272	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
49	44, 48	rspcedv 2740	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
50	36, 49	mpan2d 420	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → ((𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
51	50	rexlimdvva 2510	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
52	51	adantr 271	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
53	34, 52	mpd 13	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
54	53	expr 368	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
55		genprndu.upper	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
56	1, 2, 55	genpcuu 7176	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
57	56	alrimdv 1811	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58		breq2 3871	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑞 <Q 𝑥 ↔ 𝑞 <Q 𝑟))
59		eleq1 2157	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
60	58, 59	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑥 = 𝑟 → ((𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
61	60	cbvalv 1849	. . . . . . . . 9 ⊢ (∀𝑥(𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
62	57, 61	syl6ib 160	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63		sp 1453	. . . . . . . 8 ⊢ (∀𝑟(𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → (𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
64	62, 63	syl6 33	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
65	64	impd 252	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
66	65	ancomsd 266	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
67	66	ad2antrr 473	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
68	67	rexlimdva 2502	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
69	54, 68	impbid 128	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
70	69	ralrimiva 2458	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927  ∀wal 1294   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  {crab 2374  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690   ↦ cmpt2 5692  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936   <_Q cltq 6941  Pcnp 6947

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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-mi 6962  df-lti 6963  df-enq 7003  df-nqqs 7004  df-ltnqqs 7009  df-inp 7122

This theorem is referenced by:  addclpr  7193  mulclpr  7228