Theorem aptiprleml 7723

Description: Lemma for aptipr 7725. (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 7559	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7572	. . . . . . 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 7493	. . . . . . 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 528	. . . . . . . . 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 529	. . . . . . . 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 7559	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
11		prarloc2 7588	. . . . . . . . 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 529	. . . . . . . . . 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 7565	. . . . . . . . . . 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 531	. . . . . . . . . . 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 7565	. . . . . . . . . . 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 7480	. . . . . . . . 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 535	. . . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . 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 7484	. . . . . . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . . . 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 6097	. . . . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . . . . 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 7568	. . . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . 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 2259	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd ‘𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵)))
50		eleq1 2259	. . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . 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 7590	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . 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 621	. . . . . . . . . 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 2259	. . . . . . . . . 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 7568	. . . . . . . . . . 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 1315	. . . . . . . 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 2619	. . . . . 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 2619	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
71	4, 70	rexlimddv 2619	. . . 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 1196	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
74	73	ssrdv 3190	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 979   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157  ⟨cop 3626   class class class wbr 4034  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206  Qcnq 7364   +_Q cplq 7366   <_Q cltq 7369  Pcnp 7375  <_P cltp 7379

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iltp 7554

This theorem is referenced by:  aptipr  7725