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Theorem genprndu 7028
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 7019 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏)))
4 r2ex 2394 . . . . . . . . 9 (∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
53, 4syl6bb 194 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
65biimpa 290 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
76adantrl 462 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8 prop 6981 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnminu 6995 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
108, 9sylan 277 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
11 prop 6981 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnminu 6995 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1311, 12sylan 277 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1410, 13anim12i 331 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (2nd𝐴)) ∧ (𝐵P𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1514an4s 553 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
16 reeanv 2532 . . . . . . . . . . . . 13 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) ↔ (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1715, 16sylibr 132 . . . . . . . . . . . 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 7016 . . . . . . . . . . . . . 14 ((𝑐 <Q 𝑎𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2120reximi 2466 . . . . . . . . . . . . 13 (∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2221reximi 2466 . . . . . . . . . . . 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 463 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25 breq2 3826 . . . . . . . . . . . . . 14 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
2625biimprd 156 . . . . . . . . . . . . 13 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
2726reximdv 2470 . . . . . . . . . . . 12 (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2827reximdv 2470 . . . . . . . . . . 11 (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2928ad2antll 475 . . . . . . . . . 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 113 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3231exlimdvv 1822 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3332adantr 270 . . . . . 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 7021 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
3635imp 122 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37 elprnqu 6988 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
388, 37sylan 277 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
39 elprnqu 6988 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4011, 39sylan 277 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4138, 40anim12i 331 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (2nd𝐴)) ∧ (𝐵P𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
4241an4s 553 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
432caovcl 5758 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq1 3825 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46 eleq1 2147 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 457 . . . . . . . . . 10 (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4847adantl 271 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2719 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 419 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → ((𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2492 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5251adantr 270 . . . . 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 367 . . 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 7026 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5756alrimdv 1801 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58 breq2 3826 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑞 <Q 𝑥𝑞 <Q 𝑟))
59 eleq1 2147 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 232 . . . . . . . . . 10 (𝑥 = 𝑟 → ((𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6160cbvalv 1839 . . . . . . . . 9 (∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 159 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63 sp 1444 . . . . . . . 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 251 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6665ancomsd 265 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6766ad2antrr 472 . . . 4 ((((𝐴P𝐵P) ∧ 𝑟Q) ∧ 𝑞Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6867rexlimdva 2485 . . 3 (((𝐴P𝐵P) ∧ 𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6954, 68impbid 127 . 2 (((𝐴P𝐵P) ∧ 𝑟Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2442 1 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922  wal 1285   = wceq 1287  wex 1424  wcel 1436  wral 2355  wrex 2356  {crab 2359  cop 3434   class class class wbr 3822  cfv 4983  (class class class)co 5615  cmpt2 5617  1st c1st 5868  2nd c2nd 5869  Qcnq 6786   <Q cltq 6791  Pcnp 6797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-mi 6812  df-lti 6813  df-enq 6853  df-nqqs 6854  df-ltnqqs 6859  df-inp 6972
This theorem is referenced by:  addclpr  7043  mulclpr  7078
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