Theorem addlocpr 7344

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 7311 to both 𝐴 and 𝐵, and uses nqtri3or 7204 rather than prloc 7299 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 7216	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 <Q 𝑟 ↔ ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟))
2	1	biimpa 294	. . . . 5 ⊢ (((𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
3	2	3adant1 999	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
4		halfnqq 7218	. . . . . 6 ⊢ (𝑝 ∈ Q → ∃ℎ ∈ Q (ℎ +Q ℎ) = 𝑝)
5	4	ad2antrl 481	. . . . 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 7283	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 7311	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
8	6, 7	sylan 281	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
9	8	adantlr 468	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ℎ ∈ Q) → ∃𝑑 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑑 +Q ℎ))
10	9	3ad2antl1 1143	. . . . . . 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 500	. . . . . 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 7283	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prarloc 7311	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
14	12, 13	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
15	14	adantll 467	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ℎ ∈ Q) → ∃𝑒 ∈ (1st ‘𝐵)∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q (𝑒 +Q ℎ))
16	15	3ad2antl1 1143	. . . . . . . . . . 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 500	. . . . . . . . . 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 274	. . . . . . . . 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 1020	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 111	. . . . . . . . . . . 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 113	. . . . . . . . . . . 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 1022	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 524	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 525	. . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ +Q ℎ) = 𝑝 → (𝑞 +Q (ℎ +Q ℎ)) = (𝑞 +Q 𝑝))
29	28	eqeq1d 2148	. . . . . . . . . . . . . . 15 ⊢ ((ℎ +Q ℎ) = 𝑝 → ((𝑞 +Q (ℎ +Q ℎ)) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
30	29	ad2antll 482	. . . . . . . . . . . . . 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 166	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 526	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 527	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 525	. . . . . . . . . . . 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 526	. . . . . . . . . . . 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 527	. . . . . . . . . . . 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 521	. . . . . . . . . . . 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 7343	. . . . . . . . . . 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 372	. . . . . . . . . 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 2557	. . . . . . . . 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 372	. . . . . . 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 2557	. . . . . 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 2554	. . . 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 2554	. . 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 1183	. 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 2514	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 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   +_Q cplq 7090   <_Q cltq 7093  Pcnp 7099   +_P cpp 7101

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276

This theorem is referenced by:  addclpr  7345