Theorem addlocpr 6518

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 6485 to both A and B, and uses nqtri3or 6380 rather than prloc 6473 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 6391	. . . . . 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 921	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q) ∧ 𝑞 <Q 𝑟) → ∃𝑝 ∈ Q (𝑞 +Q 𝑝) = 𝑟)
4		halfnqq 6393	. . . . . 6 ⊢ (𝑝 ∈ Q → ∃ℎ ∈ Q (ℎ +Q ℎ) = 𝑝)
5	4	ad2antrl 459	. . . . 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 6457	. . . . . . . . . 10 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
7		prarloc 6485	. . . . . . . . . 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 1065	. . . . . . 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 478	. . . . . 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 6457	. . . . . . . . . . . . . 14 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
13		prarloc 6485	. . . . . . . . . . . . . 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 1065	. . . . . . . . . . 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 478	. . . . . . . . . 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 942	. . . . . . . . . . . . . 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 944	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 5463	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ +Q ℎ) = 𝑝 → (𝑞 +Q (ℎ +Q ℎ)) = (𝑞 +Q 𝑝))
29	28	eqeq1d 2045	. . . . . . . . . . . . . . 15 ⊢ ((ℎ +Q ℎ) = 𝑝 → ((𝑞 +Q (ℎ +Q ℎ)) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
30	29	ad2antll 460	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 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))) → u <Q (𝑑 +Q ℎ))
38		simprll 489	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6517	. . . . . . . . . . 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 2434	. . . . . . . . 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 2434	. . . . . 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 2431	. . . 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 2431	. . 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 1105	. 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 2395	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 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264   +_Q cplq 6266   <_Q cltq 6269  Pcnp 6275   +_P cpp 6277

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450

This theorem is referenced by:  addclpr  6519