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Theorem genprndu 6678
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 6669 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏)))
4 r2ex 2361 . . . . . . . . 9 (∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
53, 4syl6bb 189 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
65biimpa 284 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
76adantrl 455 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8 prop 6631 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnminu 6645 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
108, 9sylan 271 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
11 prop 6631 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnminu 6645 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1311, 12sylan 271 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1410, 13anim12i 325 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (2nd𝐴)) ∧ (𝐵P𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1514an4s 530 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
16 reeanv 2496 . . . . . . . . . . . . 13 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) ↔ (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1715, 16sylibr 141 . . . . . . . . . . . 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 6666 . . . . . . . . . . . . . 14 ((𝑐 <Q 𝑎𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2120reximi 2433 . . . . . . . . . . . . 13 (∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2221reximi 2433 . . . . . . . . . . . 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 456 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25 breq2 3796 . . . . . . . . . . . . . 14 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
2625biimprd 151 . . . . . . . . . . . . 13 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
2726reximdv 2437 . . . . . . . . . . . 12 (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2827reximdv 2437 . . . . . . . . . . 11 (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2928ad2antll 468 . . . . . . . . . 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 112 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3231exlimdvv 1793 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3332adantr 265 . . . . . 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 6671 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
3635imp 119 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37 elprnqu 6638 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
388, 37sylan 271 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
39 elprnqu 6638 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4011, 39sylan 271 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4138, 40anim12i 325 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (2nd𝐴)) ∧ (𝐵P𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
4241an4s 530 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
432caovcl 5683 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq1 3795 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46 eleq1 2116 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 450 . . . . . . . . . 10 (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4847adantl 266 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2677 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 412 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → ((𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2457 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5251adantr 265 . . . . 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 361 . . 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 6676 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5756alrimdv 1772 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58 breq2 3796 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑞 <Q 𝑥𝑞 <Q 𝑟))
59 eleq1 2116 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 227 . . . . . . . . . 10 (𝑥 = 𝑟 → ((𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6160cbvalv 1810 . . . . . . . . 9 (∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 154 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63 sp 1417 . . . . . . . 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 246 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6665ancomsd 260 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6766ad2antrr 465 . . . 4 ((((𝐴P𝐵P) ∧ 𝑟Q) ∧ 𝑞Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6867rexlimdva 2450 . . 3 (((𝐴P𝐵P) ∧ 𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6954, 68impbid 124 . 2 (((𝐴P𝐵P) ∧ 𝑟Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2409 1 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896  wal 1257   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  {crab 2327  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  cmpt2 5542  1st c1st 5793  2nd c2nd 5794  Qcnq 6436   <Q cltq 6441  Pcnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-enq 6503  df-nqqs 6504  df-ltnqqs 6509  df-inp 6622
This theorem is referenced by:  addclpr  6693  mulclpr  6728
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