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Theorem addlocpr 7308
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 7275 to both 𝐴 and 𝐵, and uses nqtri3or 7168 rather than prloc 7263 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 7180 . . . . . 6 ((𝑞Q𝑟Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 292 . . . . 5 (((𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 982 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 7182 . . . . . 6 (𝑝Q → ∃Q ( +Q ) = 𝑝)
54ad2antrl 479 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃Q ( +Q ) = 𝑝)
6 prop 7247 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 7275 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
86, 7sylan 279 . . . . . . . . 9 ((𝐴PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
98adantlr 466 . . . . . . . 8 (((𝐴P𝐵P) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1093ad2antl1 1126 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1110ad2ant2r 498 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
12 prop 7247 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prarloc 7275 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1412, 13sylan 279 . . . . . . . . . . . . 13 ((𝐵PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1514adantll 465 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1126 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 498 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1817adantr 272 . . . . . . . . 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 1003 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝐴P𝐵P))
2019ad2antrr 477 . . . . . . . . . . . . 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 111 . . . . . . . . . . . 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 113 . . . . . . . . . . . 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 1005 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 477 . . . . . . . . . . . 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 507 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → Q)
2625adantr 272 . . . . . . . . . . . 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 508 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5748 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2124 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 480 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3127, 30mpbird 166 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q ( +Q )) = 𝑟)
3231ad2antrr 477 . . . . . . . . . . . 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 509 . . . . . . . . . . . . 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 272 . . . . . . . . . . . 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 510 . . . . . . . . . . . . 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 272 . . . . . . . . . . . 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 508 . . . . . . . . . . . 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 509 . . . . . . . . . . . 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 510 . . . . . . . . . . . 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 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 ))) → 𝑡 <Q (𝑒 +Q ))
4121, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40addlocprlem 7307 . . . . . . . . . . 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 370 . . . . . . . . . 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 2532 . . . . . . . . 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 370 . . . . . . 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 2532 . . . . . 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 2529 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
493, 48rexlimddv 2529 . . 3 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
50493expia 1166 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
5150ralrimivva 2489 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 103  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  wral 2391  wrex 2392  cop 3498   class class class wbr 3897  cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  Qcnq 7052   +Q cplq 7054   <Q cltq 7057  Pcnp 7063   +P cpp 7065
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240
This theorem is referenced by:  addclpr  7309
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