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Theorem genprndu 7484
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 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genprndu.ord ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
genprndu.com ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
genprndu.upper ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
Assertion
Ref Expression
genprndu ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝑤,𝑣,𝑞,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,,𝑤,𝑣,𝑞   𝑥,𝐺,𝑦,𝑧,𝑔,,𝑤,𝑣,𝑞   𝑔,𝐹,𝑞   𝐴,𝑟,𝑞,𝑣,𝑤,𝑥,𝑦,𝑧   𝐵,𝑟,𝑔,   ,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.
StepHypRef Expression
1 genpelvl.1 . . . . . . . . . 10 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2 genpelvl.2 . . . . . . . . . 10 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpelvu 7475 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏)))
4 r2ex 2490 . . . . . . . . 9 (∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
53, 4bitrdi 195 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
65biimpa 294 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
76adantrl 475 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8 prop 7437 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnminu 7451 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
108, 9sylan 281 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
11 prop 7437 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnminu 7451 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1311, 12sylan 281 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1410, 13anim12i 336 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (2nd𝐴)) ∧ (𝐵P𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1514an4s 583 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
16 reeanv 2639 . . . . . . . . . . . . 13 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) ↔ (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1715, 16sylibr 133 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏))
18 genprndu.ord . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
19 genprndu.com . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
2018, 19genplt2i 7472 . . . . . . . . . . . . . 14 ((𝑐 <Q 𝑎𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2120reximi 2567 . . . . . . . . . . . . 13 (∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2221reximi 2567 . . . . . . . . . . . 12 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2317, 22syl 14 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2423adantrr 476 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25 breq2 3993 . . . . . . . . . . . . . 14 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
2625biimprd 157 . . . . . . . . . . . . 13 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
2726reximdv 2571 . . . . . . . . . . . 12 (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2827reximdv 2571 . . . . . . . . . . 11 (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2928ad2antll 488 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3024, 29mpd 13 . . . . . . . . 9 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟)
3130ex 114 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3231exlimdvv 1890 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3332adantr 274 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
347, 33mpd 13 . . . . 5 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟)
351, 2genppreclu 7477 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
3635imp 123 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37 elprnqu 7444 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
388, 37sylan 281 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
39 elprnqu 7444 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4011, 39sylan 281 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4138, 40anim12i 336 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (2nd𝐴)) ∧ (𝐵P𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
4241an4s 583 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
432caovcl 6007 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq1 3992 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46 eleq1 2233 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 470 . . . . . . . . . 10 (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4847adantl 275 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2838 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 426 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → ((𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2595 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5251adantr 274 . . . . 5 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5334, 52mpd 13 . . . 4 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
5453expr 373 . . 3 (((𝐴P𝐵P) ∧ 𝑟Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
55 genprndu.upper . . . . . . . . . . 11 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
561, 2, 55genpcuu 7482 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5756alrimdv 1869 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58 breq2 3993 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑞 <Q 𝑥𝑞 <Q 𝑟))
59 eleq1 2233 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝑟 → ((𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6160cbvalv 1910 . . . . . . . . 9 (∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 160 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63 sp 1504 . . . . . . . 8 (∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6462, 63syl6 33 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6564impd 252 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6665ancomsd 267 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6766ad2antrr 485 . . . 4 ((((𝐴P𝐵P) ∧ 𝑟Q) ∧ 𝑞Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6867rexlimdva 2587 . . 3 (((𝐴P𝐵P) ∧ 𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6954, 68impbid 128 . 2 (((𝐴P𝐵P) ∧ 𝑟Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2543 1 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973  wal 1346   = wceq 1348  wex 1485  wcel 2141  wral 2448  wrex 2449  {crab 2452  cop 3586   class class class wbr 3989  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117  2nd c2nd 6118  Qcnq 7242   <Q cltq 7247  Pcnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-enq 7309  df-nqqs 7310  df-ltnqqs 7315  df-inp 7428
This theorem is referenced by:  addclpr  7499  mulclpr  7534
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