Theorem aptiprleml 7859

Description: Lemma for aptipr 7861. (Contributed by Jim Kingdon, 28-Jan-2020.)

Assertion

Ref	Expression
aptiprleml	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Proof of Theorem aptiprleml

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

Step	Hyp	Ref	Expression
1		prop 7695	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7708	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
3	1, 2	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
4	3	ad2ant2rl 511	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
5		ltexnqi 7629	. . . . . . 7 ⊢ (𝑥 <Q 𝑠 → ∃𝑡 ∈ Q (𝑥 +Q 𝑡) = 𝑠)
6	5	ad2antll 491	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → ∃𝑡 ∈ Q (𝑥 +Q 𝑡) = 𝑠)
7		simplr 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐵 ∈ P)
8	7	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝐵 ∈ P)
9		simprl 531	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑡 ∈ Q)
10		prop 7695	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
11		prarloc2 7724	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
12	10, 11	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
13	8, 9, 12	syl2anc 411	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
14	8	adantr 276	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝐵 ∈ P)
15		simprl 531	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑢 ∈ (1st ‘𝐵))
16		elprnql 7701	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
17	10, 16	sylan 283	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
18	14, 15, 17	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑢 ∈ Q)
19		simpll 527	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐴 ∈ P)
20	19	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝐴 ∈ P)
21		simprr 533	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
22	21	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑥 ∈ (1st ‘𝐴))
23		elprnql 7701	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	1, 23	sylan 283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	20, 22, 24	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑥 ∈ Q)
26		nqtri3or 7616	. . . . . . . . 9 ⊢ ((𝑢 ∈ Q ∧ 𝑥 ∈ Q) → (𝑢 <Q 𝑥 ∨ 𝑢 = 𝑥 ∨ 𝑥 <Q 𝑢))
27	18, 25, 26	syl2anc 411	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 <Q 𝑥 ∨ 𝑢 = 𝑥 ∨ 𝑥 <Q 𝑢))
28	18	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑢 ∈ Q)
29		simplrl 537	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑡 ∈ Q)
30	29	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑡 ∈ Q)
31		addclnq 7595	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
32	28, 30, 31	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → (𝑢 +Q 𝑡) ∈ Q)
33		ltanqg 7620	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
34	33	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
35		addcomnqg 7601	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
36	35	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
37	34, 18, 25, 29, 36	caovord2d 6192	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 <Q 𝑥 ↔ (𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡)))
38		simplrr 538	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑥 +Q 𝑡) = 𝑠)
39		simprl 531	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st ‘𝐴))
40	39	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑠 ∈ (1st ‘𝐴))
41	38, 40	eqeltrd 2308	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑥 +Q 𝑡) ∈ (1st ‘𝐴))
42		prcdnql 7704	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st ‘𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
43	1, 42	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st ‘𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
44	20, 41, 43	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
45	37, 44	sylbid 150	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 <Q 𝑥 → (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
46		simprr 533	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
47	45, 46	jctild 316	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 <Q 𝑥 → ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))))
48	47	imp 124	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
49		eleq1 2294	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd ‘𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵)))
50		eleq1 2294	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st ‘𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
51	49, 50	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))))
52	51	rspcev 2910	. . . . . . . . . . . . 13 ⊢ (((𝑢 +Q 𝑡) ∈ Q ∧ ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))) → ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)))
53	32, 48, 52	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)))
54		ltdfpr 7726	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴))))
55	14, 20, 54	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝐵<P 𝐴 ↔ ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴))))
56	55	adantr 276	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → (𝐵<P 𝐴 ↔ ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴))))
57	53, 56	mpbird 167	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → 𝐵<P 𝐴)
58		simplrl 537	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
59	58	ad3antrrr 492	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → ¬ 𝐵<P 𝐴)
60	57, 59	pm2.21dd 625	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑥 ∈ (1st ‘𝐵))
61	60	ex 115	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 <Q 𝑥 → 𝑥 ∈ (1st ‘𝐵)))
62		eleq1 2294	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ (1st ‘𝐵) ↔ 𝑥 ∈ (1st ‘𝐵)))
63	15, 62	syl5ibcom 155	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑢 = 𝑥 → 𝑥 ∈ (1st ‘𝐵)))
64		prcdnql 7704	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
65	10, 64	sylan 283	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
66	14, 15, 65	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
67	61, 63, 66	3jaod 1340	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → ((𝑢 <Q 𝑥 ∨ 𝑢 = 𝑥 ∨ 𝑥 <Q 𝑢) → 𝑥 ∈ (1st ‘𝐵)))
68	27, 67	mpd 13	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))) → 𝑥 ∈ (1st ‘𝐵))
69	13, 68	rexlimddv 2655	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡 ∈ Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
70	6, 69	rexlimddv 2655	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
71	4, 70	rexlimddv 2655	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐵))
72	71	expr 375	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
73	72	3impa 1220	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
74	73	ssrdv 3233	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1003   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∃wrex 2511   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  2^nd c2nd 6302  Qcnq 7500   +_Q cplq 7502   <_Q cltq 7505  Pcnp 7511  <_P cltp 7515

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-iltp 7690

This theorem is referenced by:  aptipr  7861