Theorem addlocpr 6378

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 6344 to both A and B, and uses nqtri3or 6242 rather than prloc 6332 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	⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (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 ℎ 𝑝 𝑡 u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqq 6253	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟))
2	1	biimpa 280	. . . . 5 ⊢ (((𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
3	2	3adant1 904	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
4		halfnqq 6254	. . . . . 6 ⊢ (𝑝 ∈ Q → ∃ℎ ∈ Q (ℎ +Q ℎ) = 𝑝)
5	4	ad2antrl 460	. . . . 5 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → ∃ℎ ∈ Q (ℎ +Q ℎ) = 𝑝)
6		prop 6316	. . . . . . . . . 10 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
7		prarloc 6344	. . . . . . . . . 10 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ))
8	6, 7	sylan 267	. . . . . . . . 9 ⊢ ((A ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ))
9	8	adantlr 446	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P) ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ))
10	9	3ad2antl1 1048	. . . . . . 7 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ))
11	10	ad2ant2r 463	. . . . . 6 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → ∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ))
12		prop 6316	. . . . . . . . . . . . . 14 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
13		prarloc 6344	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
14	12, 13	sylan 267	. . . . . . . . . . . . 13 ⊢ ((B ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
15	14	adantll 445	. . . . . . . . . . . 12 ⊢ (((A ∈ P ∧ B ∈ P) ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
16	15	3ad2antl1 1048	. . . . . . . . . . 11 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
17	16	ad2ant2r 463	. . . . . . . . . 10 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
18	17	adantr 261	. . . . . . . . 9 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → ∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ))
19		simpll1 925	. . . . . . . . . . . . . 14 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → (A ∈ P ∧ B ∈ P))
20	19	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → (A ∈ P ∧ B ∈ P))
21	20	simpld 105	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → A ∈ P)
22	20	simprd 107	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → B ∈ P)
23		simpll3 927	. . . . . . . . . . . . 13 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → 𝑞 <Q 𝑟)
24	23	ad2antrr 457	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → 𝑞 <Q 𝑟)
25		simplrl 472	. . . . . . . . . . . . 13 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → ℎ ∈ Q)
26	25	adantr 261	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → ℎ ∈ Q)
27		simplrr 473	. . . . . . . . . . . . . 14 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28		oveq2 5433	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ +Q ℎ) = 𝑝 → (𝑞 +Q (ℎ +Q ℎ)) = (𝑞 +Q 𝑝))
29	28	eqeq1d 2022	. . . . . . . . . . . . . . 15 ⊢ ((ℎ +Q ℎ) = 𝑝 → ((𝑞 +Q (ℎ +Q ℎ)) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
30	29	ad2antll 461	. . . . . . . . . . . . . 14 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → ((𝑞 +Q (ℎ +Q ℎ)) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
31	27, 30	mpbird 156	. . . . . . . . . . . . 13 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → (𝑞 +Q (ℎ +Q ℎ)) = 𝑟)
32	31	ad2antrr 457	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → (𝑞 +Q (ℎ +Q ℎ)) = 𝑟)
33		simprll 474	. . . . . . . . . . . . 13 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → 𝑑 ∈ (1st ‘A))
34	33	adantr 261	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → 𝑑 ∈ (1st ‘A))
35		simprlr 475	. . . . . . . . . . . . 13 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → u ∈ (2nd ‘A))
36	35	adantr 261	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → u ∈ (2nd ‘A))
37		simplrr 473	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → u <Q (𝑑 +Q ℎ))
38		simprll 474	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → 𝑒 ∈ (1st ‘B))
39		simprlr 475	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → 𝑡 ∈ (2nd ‘B))
40		simprr 469	. . . . . . . . . . . 12 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → 𝑡 <Q (𝑒 +Q ℎ))
41	21, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40	addlocprlem 6377	. . . . . . . . . . 11 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ ((𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B)) ∧ 𝑡 <Q (𝑒 +Q ℎ))) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B))))
42	41	expr 357	. . . . . . . . . 10 ⊢ (((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) ∧ (𝑒 ∈ (1st ‘B) ∧ 𝑡 ∈ (2nd ‘B))) → (𝑡 <Q (𝑒 +Q ℎ) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))
43	42	rexlimdvva 2410	. . . . . . . . 9 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → (∃𝑒 ∈ (1st ‘B)∃𝑡 ∈ (2nd ‘B)𝑡 <Q (𝑒 +Q ℎ) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))
44	18, 43	mpd 13	. . . . . . . 8 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ ((𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A)) ∧ u <Q (𝑑 +Q ℎ))) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B))))
45	44	expr 357	. . . . . . 7 ⊢ ((((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) ∧ (𝑑 ∈ (1st ‘A) ∧ u ∈ (2nd ‘A))) → (u <Q (𝑑 +Q ℎ) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))
46	45	rexlimdvva 2410	. . . . . 6 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → (∃𝑑 ∈ (1st ‘A)∃u ∈ (2nd ‘A)u <Q (𝑑 +Q ℎ) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))
47	11, 46	mpd 13	. . . . 5 ⊢ (((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) ∧ (ℎ ∈ Q ∧ (ℎ +Q ℎ) = 𝑝)) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B))))
48	5, 47	rexlimddv 2407	. . . 4 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) ∧ (𝑝 ∈ Q ∧ (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B))))
49	3, 48	rexlimddv 2407	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B))))
50	49	3expia 1087	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))
51	50	ralrimivva 2371	1 ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(A +P B)) ∨ 𝑟 ∈ (2nd ‘(A +P B)))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 613   ∧ w3a 867   = wceq 1224   ∈ wcel 1367  ∀wral 2276  ∃wrex 2277  ⟨cop 3343   class class class wbr 3728  ‘cfv 4818  (class class class)co 5425  1^st c1st 5677  2^nd c2nd 5678  Qcnq 6127   +_Q cplq 6129   <_Q cltq 6132  Pcnp 6138   +_P cpp 6140

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-13 1378  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-coll 3836  ax-sep 3839  ax-nul 3847  ax-pow 3891  ax-pr 3908  ax-un 4109  ax-setind 4193  ax-iinf 4227

This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 868  df-3an 869  df-tru 1227  df-fal 1230  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ne 2180  df-ral 2281  df-rex 2282  df-reu 2283  df-rab 2285  df-v 2529  df-sbc 2734  df-csb 2822  df-dif 2889  df-un 2891  df-in 2893  df-ss 2900  df-nul 3194  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-int 3580  df-iun 3623  df-br 3729  df-opab 3783  df-mpt 3784  df-tr 3819  df-eprel 3990  df-id 3994  df-po 3997  df-iso 3998  df-iord 4042  df-on 4044  df-suc 4047  df-iom 4230  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-iota 4783  df-fun 4820  df-fn 4821  df-f 4822  df-f1 4823  df-fo 4824  df-f1o 4825  df-fv 4826  df-ov 5428  df-oprab 5429  df-mpt2 5430  df-1st 5679  df-2nd 5680  df-recs 5831  df-irdg 5867  df-1o 5905  df-2o 5906  df-oadd 5909  df-omul 5910  df-er 6006  df-ec 6008  df-qs 6012  df-ni 6151  df-pli 6152  df-mi 6153  df-lti 6154  df-plpq 6190  df-mpq 6191  df-enq 6193  df-nqqs 6194  df-plqqs 6195  df-mqqs 6196  df-1nqqs 6197  df-rq 6198  df-ltnqqs 6199  df-enq0 6266  df-nq0 6267  df-0nq0 6268  df-plq0 6269  df-mq0 6270  df-inp 6307  df-iplp 6309

This theorem is referenced by:  addclpr  6379