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Theorem genprndl 6983
Description: The lower 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)
genprndl.ord ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
genprndl.com ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
genprndl.lower ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
Assertion
Ref Expression
genprndl ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(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 genprndl
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, 2genpelvl 6974 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2392 . . . . . . . . 9 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 194 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
65biimpa 290 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
76adantrl 462 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
8 prop 6937 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnmaxl 6950 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
108, 9sylan 277 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
11 prop 6937 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 6950 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1311, 12sylan 277 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1410, 13anim12i 331 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (1st𝐴)) ∧ (𝐵P𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1514an4s 553 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
16 reeanv 2529 . . . . . . . . . . . . 13 (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) ↔ (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1715, 16sylibr 132 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ (𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑))
18 genprndl.ord . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
19 genprndl.com . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
2018, 19genplt2i 6972 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2464 . . . . . . . . . . . . 13 (∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) → ∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2464 . . . . . . . . . . . 12 (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2317, 22syl 14 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2423adantrr 463 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 3814 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 156 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2468 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2468 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 475 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3024, 29mpd 13 . . . . . . . . 9 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑))
3130ex 113 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3231exlimdvv 1820 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3332adantr 270 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
347, 33mpd 13 . . . . 5 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑))
351, 2genpprecll 6976 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵)) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
3635imp 122 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))
37 elprnql 6943 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (1st𝐴)) → 𝑐Q)
388, 37sylan 277 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (1st𝐴)) → 𝑐Q)
39 elprnql 6943 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4011, 39sylan 277 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4138, 40anim12i 331 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (1st𝐴)) ∧ (𝐵P𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
4241an4s 553 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
432caovcl 5734 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq2 3815 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2145 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 457 . . . . . . . . . 10 (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4847adantl 271 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) ∧ 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2716 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → ((𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 419 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2490 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5251adantr 270 . . . . 5 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5334, 52mpd 13 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))))
5453expr 367 . . 3 (((𝐴P𝐵P) ∧ 𝑞Q) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
55 genprndl.lower . . . . . . . . . . 11 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
561, 2, 55genpcdl 6981 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
5756alrimdv 1799 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
58 breq1 3814 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2145 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 232 . . . . . . . . . 10 (𝑥 = 𝑞 → ((𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
6160cbvalv 1837 . . . . . . . . 9 (∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 159 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
63 sp 1442 . . . . . . . 8 (∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵))) → (𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6462, 63syl6 33 . . . . . . 7 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
6564impd 251 . . . . . 6 ((𝐴P𝐵P) → ((𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6665ancomsd 265 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6766ad2antrr 472 . . . 4 ((((𝐴P𝐵P) ∧ 𝑞Q) ∧ 𝑟Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6867rexlimdva 2483 . . 3 (((𝐴P𝐵P) ∧ 𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6954, 68impbid 127 . 2 (((𝐴P𝐵P) ∧ 𝑞Q) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2440 1 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920  wal 1283   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  {crab 2357  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  cmpt2 5593  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   <Q cltq 6747  Pcnp 6753
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-mi 6768  df-lti 6769  df-enq 6809  df-nqqs 6810  df-ltnqqs 6815  df-inp 6928
This theorem is referenced by:  addclpr  6999  mulclpr  7034
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