Theorem addlocpr 7596

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 7563 to both 𝐴 and 𝐵, and uses nqtri3or 7456 rather than prloc 7551 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 7468	. . . . . 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 1017	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
4		halfnqq 7470	. . . . . 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 7535	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 7563	. . . . . . . . . 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 1161	. . . . . . 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 7535	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prarloc 7563	. . . . . . . . . . . . . 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 1161	. . . . . . . . . . 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 1038	. . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . 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 5926	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ +Q ℎ) = 𝑝 → (𝑞 +Q (ℎ +Q ℎ)) = (𝑞 +Q 𝑝))
29	28	eqeq1d 2202	. . . . . . . . . . . . . . 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 7595	. . . . . . . . . . 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 2619	. . . . . . . . 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 2619	. . . . . 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 2616	. . . 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 2616	. . 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 1207	. 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 2576	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 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ⟨cop 3621   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340   +_Q cplq 7342   <_Q cltq 7345  Pcnp 7351   +_P cpp 7353

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528

This theorem is referenced by:  addclpr  7597