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Theorem addlocpr 7498
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 7465 to both 𝐴 and 𝐵, and uses nqtri3or 7358 rather than prloc 7453 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 7370 . . . . . 6 ((𝑞Q𝑟Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 294 . . . . 5 (((𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 1010 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 7372 . . . . . 6 (𝑝Q → ∃Q ( +Q ) = 𝑝)
54ad2antrl 487 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃Q ( +Q ) = 𝑝)
6 prop 7437 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 7465 . . . . . . . . . 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 474 . . . . . . . 8 (((𝐴P𝐵P) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1093ad2antl1 1154 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1110ad2ant2r 506 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
12 prop 7437 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prarloc 7465 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1154 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 506 . . . . . . . . . 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 1031 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝐴P𝐵P))
2019ad2antrr 485 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 485 . . . . . . . . . . . 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 530 . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5861 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2179 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 488 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . 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 532 . . . . . . . . . . . . 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 533 . . . . . . . . . . . . 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 531 . . . . . . . . . . . 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 532 . . . . . . . . . . . 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 533 . . . . . . . . . . . 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 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 ))) → 𝑡 <Q (𝑒 +Q ))
4121, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40addlocprlem 7497 . . . . . . . . . . 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 2595 . . . . . . . . 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 2595 . . . . . 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 2592 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
493, 48rexlimddv 2592 . . 3 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
50493expia 1200 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
5150ralrimivva 2552 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 703  w3a 973   = wceq 1348  wcel 2141  wral 2448  wrex 2449  cop 3586   class class class wbr 3989  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  Qcnq 7242   +Q cplq 7244   <Q cltq 7247  Pcnp 7253   +P cpp 7255
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430
This theorem is referenced by:  addclpr  7499
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