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Theorem addlocpr 7456
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 7423 to both 𝐴 and 𝐵, and uses nqtri3or 7316 rather than prloc 7411 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 7328 . . . . . 6 ((𝑞Q𝑟Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 294 . . . . 5 (((𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 1000 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 7330 . . . . . 6 (𝑝Q → ∃Q ( +Q ) = 𝑝)
54ad2antrl 482 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃Q ( +Q ) = 𝑝)
6 prop 7395 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 7423 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
86, 7sylan 281 . . . . . . . . 9 ((𝐴PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
98adantlr 469 . . . . . . . 8 (((𝐴P𝐵P) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1093ad2antl1 1144 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1110ad2ant2r 501 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
12 prop 7395 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prarloc 7423 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1412, 13sylan 281 . . . . . . . . . . . . 13 ((𝐵PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1514adantll 468 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1144 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 501 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1817adantr 274 . . . . . . . . 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 1021 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝐴P𝐵P))
2019ad2antrr 480 . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 480 . . . . . . . . . . . 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 525 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → Q)
2625adantr 274 . . . . . . . . . . . 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 526 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5832 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2166 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 483 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 527 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 528 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 526 . . . . . . . . . . . 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 527 . . . . . . . . . . . 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 528 . . . . . . . . . . . 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 522 . . . . . . . . . . . 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 7455 . . . . . . . . . . 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 373 . . . . . . . . . 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 2582 . . . . . . . . 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 373 . . . . . . 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 2582 . . . . . 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 2579 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
493, 48rexlimddv 2579 . . 3 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
50493expia 1187 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
5150ralrimivva 2539 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 698  w3a 963   = wceq 1335  wcel 2128  wral 2435  wrex 2436  cop 3563   class class class wbr 3965  cfv 5170  (class class class)co 5824  1st c1st 6086  2nd c2nd 6087  Qcnq 7200   +Q cplq 7202   <Q cltq 7205  Pcnp 7211   +P cpp 7213
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388
This theorem is referenced by:  addclpr  7457
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