Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addlocpr GIF version

 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 6658 to both 𝐴 and 𝐵, and uses nqtri3or 6551 rather than prloc 6646 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

Dummy variables 𝑑 𝑒 𝑝 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqq 6563 . . . . . 6 ((𝑞Q𝑟Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 284 . . . . 5 (((𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 933 . . . 4 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 6565 . . . . . 6 (𝑝Q → ∃Q ( +Q ) = 𝑝)
54ad2antrl 467 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃Q ( +Q ) = 𝑝)
6 prop 6630 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prarloc 6658 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
86, 7sylan 271 . . . . . . . . 9 ((𝐴PQ) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
98adantlr 454 . . . . . . . 8 (((𝐴P𝐵P) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1093ad2antl1 1077 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
1110ad2ant2r 486 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑑 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑑 +Q ))
12 prop 6630 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
13 prarloc 6658 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1412, 13sylan 271 . . . . . . . . . . . . 13 ((𝐵PQ) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1514adantll 453 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1077 . . . . . . . . . . 11 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ Q) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 486 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ∃𝑒 ∈ (1st𝐵)∃𝑡 ∈ (2nd𝐵)𝑡 <Q (𝑒 +Q ))
1817adantr 265 . . . . . . . . 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 954 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝐴P𝐵P))
2019ad2antrr 465 . . . . . . . . . . . . 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 109 . . . . . . . . . . . 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 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 ))) → 𝐵P)
23 simpll3 956 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 465 . . . . . . . . . . . 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 495 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) ∧ ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)) ∧ 𝑢 <Q (𝑑 +Q ))) → Q)
2625adantr 265 . . . . . . . . . . . 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 496 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5547 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2064 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 468 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3127, 30mpbird 160 . . . . . . . . . . . . 13 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (Q ∧ ( +Q ) = 𝑝)) → (𝑞 +Q ( +Q )) = 𝑟)
3231ad2antrr 465 . . . . . . . . . . . 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 497 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 496 . . . . . . . . . . . 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 497 . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 6690 . . . . . . . . . . 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 361 . . . . . . . . . 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 2457 . . . . . . . . 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 361 . . . . . . 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 2457 . . . . . 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 2454 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
493, 48rexlimddv 2454 . . 3 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))
50493expia 1117 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
5150ralrimivva 2418 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 101   ↔ wb 102   ∨ wo 639   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  ⟨cop 3405   class class class wbr 3791  ‘cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437
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