Theorem aptiprlemu 8001

Description: Lemma for aptipr 8002. (Contributed by Jim Kingdon, 28-Jan-2020.)

Assertion

Ref	Expression
aptiprlemu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))

Proof of Theorem aptiprlemu

Dummy variables 𝑓 𝑔 ℎ 𝑠 𝑡 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7836	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7850	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q 𝑥)
3	1, 2	sylan 283	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q 𝑥)
4	3	3ad2antl2 1191	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q 𝑥)
5		simprr 537	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 <Q 𝑥)
6		ltexnqi 7770	. . . . . 6 ⊢ (𝑠 <Q 𝑥 → ∃𝑡 ∈ Q (𝑠 +Q 𝑡) = 𝑥)
7	5, 6	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → ∃𝑡 ∈ Q (𝑠 +Q 𝑡) = 𝑥)
8		simpl1 1031	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝐴 ∈ P)
9	8	ad2antrr 492	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝐴 ∈ P)
10		simprl 535	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑡 ∈ Q)
11		prop 7836	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		prarloc2 7865	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
13	11, 12	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
14	9, 10, 13	syl2anc 415	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
15		simpl2 1032	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝐵 ∈ P)
16	15	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐵 ∈ P)
17		simpr 110	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ (2nd ‘𝐵))
18	17	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑥 ∈ (2nd ‘𝐵))
19		elprnqu 7843	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
20	1, 19	sylan 283	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
21	16, 18, 20	syl2anc 415	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑥 ∈ Q)
22	8	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐴 ∈ P)
23		simprl 535	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ (1st ‘𝐴))
24		elprnql 7842	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
25	11, 24	sylan 283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
26	22, 23, 25	syl2anc 415	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ Q)
27	10	adantr 276	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑡 ∈ Q)
28		addclnq 7736	. . . . . . . . 9 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
29	26, 27, 28	syl2anc 415	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑢 +Q 𝑡) ∈ Q)
30		nqtri3or 7757	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (𝑢 +Q 𝑡) ∈ Q) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
31	21, 29, 30	syl2anc 415	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
32	15	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → 𝐵 ∈ P)
33		simprl 535	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ (2nd ‘𝐵))
34		elprnqu 7843	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
35	1, 34	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
36	32, 33, 35	syl2anc 415	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ Q)
37	36	ad3antrrr 496	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ Q)
38	33	ad3antrrr 496	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (2nd ‘𝐵))
39		simplrr 542	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑠 +Q 𝑡) = 𝑥)
40		breq1 4131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 +Q 𝑡) = 𝑥 → ((𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡) ↔ 𝑥 <Q (𝑢 +Q 𝑡)))
41	40	biimprd 158	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 +Q 𝑡) = 𝑥 → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
42	39, 41	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
43	42	imp 124	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡))
44		ltanqg 7761	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
45	44	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
46	26	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢 ∈ Q)
47	27	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑡 ∈ Q)
48		addcomnqg 7742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
49	48	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
50	45, 37, 46, 47, 49	caovord2d 6253	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢 ↔ (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
51	43, 50	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 <Q 𝑢)
52	22	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐴 ∈ P)
53	23	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢 ∈ (1st ‘𝐴))
54		prcdnql 7845	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → (𝑠 <Q 𝑢 → 𝑠 ∈ (1st ‘𝐴)))
55	11, 54	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → (𝑠 <Q 𝑢 → 𝑠 ∈ (1st ‘𝐴)))
56	52, 53, 55	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢 → 𝑠 ∈ (1st ‘𝐴)))
57	51, 56	mpd 13	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (1st ‘𝐴))
58		rspe 2599	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝐴)))
59	37, 38, 57, 58	syl12anc 1276	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝐴)))
60	16	adantr 276	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵 ∈ P)
61		ltdfpr 7867	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝐴))))
62	60, 52, 61	syl2anc 415	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝐵<P 𝐴 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝐴))))
63	59, 62	mpbird 167	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵<P 𝐴)
64		simpll3 1069	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → ¬ 𝐵<P 𝐴)
65	64	ad3antrrr 496	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ¬ 𝐵<P 𝐴)
66	63, 65	pm2.21dd 629	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑥 ∈ (2nd ‘𝐴))
67	66	ex 115	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd ‘𝐴)))
68		simprr 537	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
69		eleq1 2301	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 +Q 𝑡) → (𝑥 ∈ (2nd ‘𝐴) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)))
70	68, 69	syl5ibrcom 157	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑥 = (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd ‘𝐴)))
71		prcunqu 7846	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
72	11, 71	sylan 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
73	22, 68, 72	syl2anc 415	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) <Q 𝑥 → 𝑥 ∈ (2nd ‘𝐴)))
74	67, 70, 73	3jaod 1345	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥) → 𝑥 ∈ (2nd ‘𝐴)))
75	31, 74	mpd 13	. . . . . 6 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑥 ∈ (2nd ‘𝐴))
76	14, 75	rexlimddv 2673	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡 ∈ Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑥 ∈ (2nd ‘𝐴))
77	7, 76	rexlimddv 2673	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑥 ∈ (2nd ‘𝐴))
78	4, 77	rexlimddv 2673	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ (2nd ‘𝐴))
79	78	ex 115	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (2nd ‘𝐵) → 𝑥 ∈ (2nd ‘𝐴)))
80	79	ssrdv 3254	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1008   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   +_Q cplq 7643   <_Q cltq 7646  Pcnp 7652  <_P cltp 7656

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-iltp 7831

This theorem is referenced by:  aptipr  8002  suplocexprlemmu  8079