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Theorem genprndu 6618
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 6609 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏)))
4 r2ex 2344 . . . . . . . . 9 (∃𝑎 ∈ (2nd𝐴)∃𝑏 ∈ (2nd𝐵)𝑟 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))))
65biimpa 280 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
76adantrl 447 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑟Q𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)))
8 prop 6571 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnminu 6585 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
108, 9sylan 267 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (2nd𝐴)) → ∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎)
11 prop 6571 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnminu 6585 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1311, 12sylan 267 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (2nd𝐵)) → ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏)
1410, 13anim12i 321 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (2nd𝐴)) ∧ (𝐵P𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1514an4s 522 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵))) → (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
16 reeanv 2479 . . . . . . . . . . . . 13 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) ↔ (∃𝑐 ∈ (2nd𝐴)𝑐 <Q 𝑎 ∧ ∃𝑑 ∈ (2nd𝐵)𝑑 <Q 𝑏))
1715, 16sylibr 137 . . . . . . . . . . . 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 6606 . . . . . . . . . . . . . 14 ((𝑐 <Q 𝑎𝑑 <Q 𝑏) → (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2120reximi 2416 . . . . . . . . . . . . 13 (∃𝑑 ∈ (2nd𝐵)(𝑐 <Q 𝑎𝑑 <Q 𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
2221reximi 2416 . . . . . . . . . . . 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 448 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏))
25 breq2 3768 . . . . . . . . . . . . . 14 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q (𝑎𝐺𝑏)))
2625biimprd 147 . . . . . . . . . . . . 13 (𝑟 = (𝑎𝐺𝑏) → ((𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → (𝑐𝐺𝑑) <Q 𝑟))
2726reximdv 2420 . . . . . . . . . . . 12 (𝑟 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2827reximdv 2420 . . . . . . . . . . 11 (𝑟 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q (𝑎𝐺𝑏) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
2928ad2antll 460 . . . . . . . . . 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 108 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3231exlimdvv 1777 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (2nd𝐴) ∧ 𝑏 ∈ (2nd𝐵)) ∧ 𝑟 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟))
3332adantr 261 . . . . . 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 6611 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
3635imp 115 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))
37 elprnqu 6578 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
388, 37sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (2nd𝐴)) → 𝑐Q)
39 elprnqu 6578 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4011, 39sylan 267 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (2nd𝐵)) → 𝑑Q)
4138, 40anim12i 321 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (2nd𝐴)) ∧ (𝐵P𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
4241an4s 522 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐Q𝑑Q))
432caovcl 5655 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq1 3767 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟 ↔ (𝑐𝐺𝑑) <Q 𝑟))
46 eleq1 2100 . . . . . . . . . . 11 (𝑞 = (𝑐𝐺𝑑) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 442 . . . . . . . . . 10 (𝑞 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4847adantl 262 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) ∧ 𝑞 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2660 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (((𝑐𝐺𝑑) <Q 𝑟 ∧ (𝑐𝐺𝑑) ∈ (2nd ‘(𝐴𝐹𝐵))) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 404 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → ((𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2440 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)(𝑐𝐺𝑑) <Q 𝑟 → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5251adantr 261 . . . . 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 357 . . 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 6616 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
5756alrimdv 1756 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
58 breq2 3768 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑞 <Q 𝑥𝑞 <Q 𝑟))
59 eleq1 2100 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 223 . . . . . . . . . 10 (𝑥 = 𝑟 → ((𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6160cbvalv 1794 . . . . . . . . 9 (∀𝑥(𝑞 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 150 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → ∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
63 sp 1401 . . . . . . . 8 (∀𝑟(𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))) → (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6462, 63syl6 29 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑟𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)))))
6564impd 242 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6665ancomsd 256 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6766ad2antrr 457 . . . 4 ((((𝐴P𝐵P) ∧ 𝑟Q) ∧ 𝑞Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6867rexlimdva 2433 . . 3 (((𝐴P𝐵P) ∧ 𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) → 𝑟 ∈ (2nd ‘(𝐴𝐹𝐵))))
6954, 68impbid 120 . 2 (((𝐴P𝐵P) ∧ 𝑟Q) → (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2392 1 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885  wal 1241   = wceq 1243  wex 1381  wcel 1393  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6376   <Q cltq 6381  Pcnp 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-mi 6402  df-lti 6403  df-enq 6443  df-nqqs 6444  df-ltnqqs 6449  df-inp 6562
This theorem is referenced by:  addclpr  6633  mulclpr  6668
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