Theorem addlocpr 7679

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 7646 to both 𝐴 and 𝐵, and uses nqtri3or 7539 rather than prloc 7634 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 7551	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟))
2	1	biimpa 296	. . . . 5 ⊢ (((𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
3	2	3adant1 1018	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
4		halfnqq 7553	. . . . . 6 ⊢ (𝑝 ∈ Q → ∃ℎ ∈ Q (ℎ +Q ℎ) = 𝑝)
5	4	ad2antrl 490	. . . . 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 7618	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 7646	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
8	6, 7	sylan 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
9	8	adantlr 477	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
10	9	3ad2antl1 1162	. . . . . . 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 509	. . . . . 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 7618	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prarloc 7646	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
14	12, 13	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
15	14	adantll 476	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
16	15	3ad2antl1 1162	. . . . . . . . . . 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 509	. . . . . . . . . 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 276	. . . . . . . . 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 1039	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 112	. . . . . . . . . . . 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 114	. . . . . . . . . . . 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 1041	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 536	. . . . . . . . . . . . . 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 5970	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ +Q ℎ) = 𝑝 → (𝑞 +Q (ℎ +Q ℎ)) = (𝑞 +Q 𝑝))
29	28	eqeq1d 2215	. . . . . . . . . . . . . . 15 ⊢ ((ℎ +Q ℎ) = 𝑝 → ((𝑞 +Q (ℎ +Q ℎ)) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
30	29	ad2antll 491	. . . . . . . . . . . . . 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 167	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 537	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 536	. . . . . . . . . . . 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 537	. . . . . . . . . . . 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 538	. . . . . . . . . . . 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 531	. . . . . . . . . . . 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 7678	. . . . . . . . . . 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 375	. . . . . . . . . 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 2632	. . . . . . . . 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 375	. . . . . . 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 2632	. . . . . 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 2629	. . . 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 2629	. . 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 1208	. 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 2589	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 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  ⟨cop 3641   class class class wbr 4054  ‘cfv 5285  (class class class)co 5962  1^st c1st 6242  2^nd c2nd 6243  Qcnq 7423   +_Q cplq 7425   <_Q cltq 7428  Pcnp 7434   +_P cpp 7436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-2o 6521  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-enq0 7567  df-nq0 7568  df-0nq0 7569  df-plq0 7570  df-mq0 7571  df-inp 7609  df-iplp 7611

This theorem is referenced by:  addclpr  7680