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Theorem genprndl 7341
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 7332 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2455 . . . . . . . . 9 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 195 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
65biimpa 294 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
76adantrl 469 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
8 prop 7295 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnmaxl 7308 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
108, 9sylan 281 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
11 prop 7295 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 7308 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1311, 12sylan 281 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1410, 13anim12i 336 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (1st𝐴)) ∧ (𝐵P𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1514an4s 577 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
16 reeanv 2600 . . . . . . . . . . . . 13 (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) ↔ (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1715, 16sylibr 133 . . . . . . . . . . . 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 7330 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2529 . . . . . . . . . . . . 13 (∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) → ∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2529 . . . . . . . . . . . 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 470 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 3932 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 157 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2533 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2533 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 482 . . . . . . . . . 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 114 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3231exlimdvv 1869 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3332adantr 274 . . . . . 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 7334 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵)) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
3635imp 123 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))
37 elprnql 7301 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (1st𝐴)) → 𝑐Q)
388, 37sylan 281 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (1st𝐴)) → 𝑐Q)
39 elprnql 7301 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4011, 39sylan 281 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4138, 40anim12i 336 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (1st𝐴)) ∧ (𝐵P𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
4241an4s 577 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
432caovcl 5925 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq2 3933 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2202 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 464 . . . . . . . . . 10 (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4847adantl 275 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) ∧ 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2793 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → ((𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 424 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2557 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5251adantr 274 . . . . 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 372 . . 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 7339 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
5756alrimdv 1848 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
58 breq1 3932 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2202 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝑞 → ((𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
6160cbvalv 1889 . . . . . . . . 9 (∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6257, 61syl6ib 160 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
63 sp 1488 . . . . . . . 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 252 . . . . . 6 ((𝐴P𝐵P) → ((𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6665ancomsd 267 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6766ad2antrr 479 . . . 4 ((((𝐴P𝐵P) ∧ 𝑞Q) ∧ 𝑟Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6867rexlimdva 2549 . . 3 (((𝐴P𝐵P) ∧ 𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6954, 68impbid 128 . 2 (((𝐴P𝐵P) ∧ 𝑞Q) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2505 1 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962  wal 1329   = wceq 1331  wex 1468  wcel 1480  wral 2416  wrex 2417  {crab 2420  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   <Q cltq 7105  Pcnp 7111
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-mi 7126  df-lti 7127  df-enq 7167  df-nqqs 7168  df-ltnqqs 7173  df-inp 7286
This theorem is referenced by:  addclpr  7357  mulclpr  7392
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