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Theorem genprndl 7550
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 7541 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2510 . . . . . . . . 9 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4bitrdi 196 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
65biimpa 296 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
76adantrl 478 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞Q𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
8 prop 7504 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 prnmaxl 7517 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
108, 9sylan 283 . . . . . . . . . . . . . . 15 ((𝐴P𝑎 ∈ (1st𝐴)) → ∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐)
11 prop 7504 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
12 prnmaxl 7517 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1311, 12sylan 283 . . . . . . . . . . . . . . 15 ((𝐵P𝑏 ∈ (1st𝐵)) → ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑)
1410, 13anim12i 338 . . . . . . . . . . . . . 14 (((𝐴P𝑎 ∈ (1st𝐴)) ∧ (𝐵P𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1514an4s 588 . . . . . . . . . . . . 13 (((𝐴P𝐵P) ∧ (𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵))) → (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
16 reeanv 2660 . . . . . . . . . . . . 13 (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) ↔ (∃𝑐 ∈ (1st𝐴)𝑎 <Q 𝑐 ∧ ∃𝑑 ∈ (1st𝐵)𝑏 <Q 𝑑))
1715, 16sylibr 134 . . . . . . . . . . . 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 7539 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2587 . . . . . . . . . . . . 13 (∃𝑑 ∈ (1st𝐵)(𝑎 <Q 𝑐𝑏 <Q 𝑑) → ∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2587 . . . . . . . . . . . 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 479 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 4021 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 158 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2591 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2591 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 491 . . . . . . . . . 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 115 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3231exlimdvv 1909 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) → ∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑)))
3332adantr 276 . . . . . 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 7543 . . . . . . . . 9 ((𝐴P𝐵P) → ((𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵)) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
3635imp 124 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))
37 elprnql 7510 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑐 ∈ (1st𝐴)) → 𝑐Q)
388, 37sylan 283 . . . . . . . . . . . 12 ((𝐴P𝑐 ∈ (1st𝐴)) → 𝑐Q)
39 elprnql 7510 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4011, 39sylan 283 . . . . . . . . . . . 12 ((𝐵P𝑑 ∈ (1st𝐵)) → 𝑑Q)
4138, 40anim12i 338 . . . . . . . . . . 11 (((𝐴P𝑐 ∈ (1st𝐴)) ∧ (𝐵P𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
4241an4s 588 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐Q𝑑Q))
432caovcl 6051 . . . . . . . . . 10 ((𝑐Q𝑑Q) → (𝑐𝐺𝑑) ∈ Q)
4442, 43syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑐𝐺𝑑) ∈ Q)
45 breq2 4022 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2252 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))))
4745, 46anbi12d 473 . . . . . . . . . 10 (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4847adantl 277 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) ∧ 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵)))))
4944, 48rspcedv 2860 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → ((𝑞 <Q (𝑐𝐺𝑑) ∧ (𝑐𝐺𝑑) ∈ (1st ‘(𝐴𝐹𝐵))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5036, 49mpan2d 428 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑐 ∈ (1st𝐴) ∧ 𝑑 ∈ (1st𝐵))) → (𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5150rexlimdvva 2615 . . . . . 6 ((𝐴P𝐵P) → (∃𝑐 ∈ (1st𝐴)∃𝑑 ∈ (1st𝐵)𝑞 <Q (𝑐𝐺𝑑) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
5251adantr 276 . . . . 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 375 . . 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 7548 . . . . . . . . . 10 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
5756alrimdv 1887 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
58 breq1 4021 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2252 . . . . . . . . . . 11 (𝑥 = 𝑞 → (𝑥 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6058, 59imbi12d 234 . . . . . . . . . 10 (𝑥 = 𝑞 → ((𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ (𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
6160cbvalv 1929 . . . . . . . . 9 (∀𝑥(𝑥 <Q 𝑟𝑥 ∈ (1st ‘(𝐴𝐹𝐵))) ↔ ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6257, 61imbitrdi 161 . . . . . . . 8 ((𝐴P𝐵P) → (𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) → ∀𝑞(𝑞 <Q 𝑟𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))))
63 sp 1522 . . . . . . . 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 254 . . . . . 6 ((𝐴P𝐵P) → ((𝑟 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6665ancomsd 269 . . . . 5 ((𝐴P𝐵P) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6766ad2antrr 488 . . . 4 ((((𝐴P𝐵P) ∧ 𝑞Q) ∧ 𝑟Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6867rexlimdva 2607 . . 3 (((𝐴P𝐵P) ∧ 𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵))) → 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
6954, 68impbid 129 . 2 (((𝐴P𝐵P) ∧ 𝑞Q) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
7069ralrimiva 2563 1 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  {crab 2472  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5896  cmpo 5898  1st c1st 6163  2nd c2nd 6164  Qcnq 7309   <Q cltq 7314  Pcnp 7320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-mi 7335  df-lti 7336  df-enq 7376  df-nqqs 7377  df-ltnqqs 7382  df-inp 7495
This theorem is referenced by:  addclpr  7566  mulclpr  7601
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