Theorem genprndu 7523

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 7514	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd ‘𝐴)∃𝑏 ∈ (2nd ‘𝐵)𝑟 = (𝑎𝐺𝑏)))
4		r2ex 2497	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (2nd ‘𝐴)∃𝑏 ∈ (2nd ‘𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
5	3, 4	bitrdi 196	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
6	5	biimpa 296	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
7	6	adantrl 478	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎∃𝑏((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8		prop 7476	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
9		prnminu 7490	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (2nd ‘𝐴)) → ∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎)
10	8, 9	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝑎 ∈ (2nd ‘𝐴)) → ∃𝑐 ∈ (2nd ‘𝐴)𝑐 <Q 𝑎)
11		prop 7476	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12		prnminu 7490	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (2nd ‘𝐵)) → ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏)
13	11, 12	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ 𝑏 ∈ (2nd ‘𝐵)) → ∃𝑑 ∈ (2nd ‘𝐵)𝑑 <Q 𝑏)
14	10, 13	anim12i 338	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 2647	. . . . . . . . . . . . 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 134	. . . . . . . . . . . 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 7511	. . . . . . . . . . . . . 14 ⊢ ((𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
21	20	reximi 2574	. . . . . . . . . . . . 13 ⊢ (∃𝑑 ∈ (2nd ‘𝐵)(𝑐 <Q 𝑎 ∧ 𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
22	21	reximi 2574	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (2nd ‘𝐴) ∧ 𝑏 ∈ (2nd ‘𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25		breq2 4009	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
26	25	biimprd 158	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
27	26	reximdv 2578	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
28	27	reximdv 2578	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)(𝑐𝐺𝑑) <Q 𝑟))
29	28	ad2antll 491	. . . . . . . . . 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 115	. . . . . . . 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 1897	. . . . . . 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 276	. . . . . 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 7516	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
36	35	imp 124	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37		elprnqu 7483	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑐 ∈ Q)
38	8, 37	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑐 ∈ Q)
39		elprnqu 7483	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑑 ∈ Q)
40	11, 39	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑑 ∈ Q)
41	38, 40	anim12i 338	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝑐 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
42	41	an4s 588	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑐 ∈ Q ∧ 𝑑 ∈ Q))
43	2	caovcl 6031	. . . . . . . . . 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 4008	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46		eleq1 2240	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
47	45, 46	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
48	47	adantl 277	. . . . . . . . 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 2847	. . . . . . . 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 428	. . . . . . 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 2602	. . . . . 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 276	. . . . 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 375	. . 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 7521	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
57	56	alrimdv 1876	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58		breq2 4009	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑞 <Q 𝑥 ↔ 𝑞 <Q 𝑟))
59		eleq1 2240	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
60	58, 59	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = 𝑟 → ((𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
61	60	cbvalv 1917	. . . . . . . . 9 ⊢ (∀𝑥(𝑞 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
62	57, 61	imbitrdi 161	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟 → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63		sp 1511	. . . . . . . 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 254	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
66	65	ancomsd 269	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
67	66	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) → ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
68	67	rexlimdva 2594	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
69	54, 68	impbid 129	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
70	69	ralrimiva 2550	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459  ⟨cop 3597   class class class wbr 4005  ‘cfv 5218  (class class class)co 5877   ∈ cmpo 5879  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   <_Q cltq 7286  Pcnp 7292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-enq 7348  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467

This theorem is referenced by:  addclpr  7538  mulclpr  7573