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Theorem addlocpr 7074
Description: Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7041 to both 𝐴 and 𝐵, and uses nqtri3or 6934 rather than prloc 7029 to decide whether 𝑞 is too big to be in the lower cut of 𝐴 +P 𝐵 (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addlocpr ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
Distinct variable groups:   𝐴,𝑞,𝑟   𝐵,𝑞,𝑟

Proof of Theorem addlocpr
Dummy variables 𝑑 𝑒 𝑝 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqq 6946 . . . . . 6 ((𝑞Q𝑟Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 290 . . . . 5 (((𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 961 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 6948 . . . . . 6 (𝑝Q → ∃Q ( +Q ) = 𝑝)
54ad2antrl 474 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃Q ( +Q ) = 𝑝)
6 prop 7013 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 7041 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
86, 7sylan 277 . . . . . . . . 9 ((𝐴PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
98adantlr 461 . . . . . . . 8 (((𝐴P𝐵P) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1093ad2antl1 1105 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1110ad2ant2r 493 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
12 prop 7013 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prarloc 7041 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1412, 13sylan 277 . . . . . . . . . . . . 13 ((𝐵PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1514adantll 460 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1105 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 493 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1817adantr 270 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
19 simpll1 982 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝐴P𝐵P))
2019ad2antrr 472 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → (𝐴P𝐵P))
2120simpld 110 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝐴P)
2220simprd 112 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝐵P)
23 simpll3 984 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 472 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑞 <Q 𝑟)
25 simplrl 502 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → Q)
2625adantr 270 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → Q)
27 simplrr 503 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5642 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2096 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 475 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3127, 30mpbird 165 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q ( +Q )) = 𝑟)
3231ad2antrr 472 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → (𝑞 +Q ( +Q )) = 𝑟)
33 simprll 504 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → 𝑑 ∈ (1st𝐴))
3433adantr 270 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑑 ∈ (1st𝐴))
35 simprlr 505 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → 𝑢 ∈ (2nd𝐴))
3635adantr 270 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑢 ∈ (2nd𝐴))
37 simplrr 503 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑢 <Q (𝑑 +Q ))
38 simprll 504 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑒 ∈ (1st𝐵))
39 simprlr 505 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑡 ∈ (2nd𝐵))
40 simprr 499 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → 𝑡 <Q (𝑒 +Q ))
4121, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40addlocprlem 7073 . . . . . . . . . . 11 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ ((𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) ∧ 𝑡 <Q (𝑒 +Q ))) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
4241expr 367 . . . . . . . . . 10 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) ∧ (𝑒 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵))) → (𝑡 <Q (𝑒 +Q ) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
4342rexlimdvva 2496 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → (∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
4418, 43mpd 13 . . . . . . . 8 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
4544expr 367 . . . . . . 7 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑢 <Q (𝑑 +Q ) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
4645rexlimdvva 2496 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
4711, 46mpd 13 . . . . 5 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
485, 47rexlimddv 2493 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
493, 48rexlimddv 2493 . . 3 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
50493expia 1145 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
5150ralrimivva 2455 1 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 664  w3a 924   = wceq 1289  wcel 1438  wral 2359  wrex 2360  cop 3444   class class class wbr 3837  cfv 5002  (class class class)co 5634  1st c1st 5891  2nd c2nd 5892  Qcnq 6818   +Q cplq 6820   <Q cltq 6823  Pcnp 6829   +P cpp 6831
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006
This theorem is referenced by:  addclpr  7075
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