Theorem aptiprleml 7601

Description: Lemma for aptipr 7603. (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 7437	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7450	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
3	1, 2	sylan 281	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
4	3	ad2ant2rl 508	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
5		ltexnqi 7371	. . . . . . 7 ⊢ (𝑥 <Q 𝑠 → ∃𝑡 ∈ Q (𝑥 +Q 𝑡) = 𝑠)
6	5	ad2antll 488	. . . . . 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 525	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐵 ∈ P)
8	7	ad2antrr 485	. . . . . . . 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 526	. . . . . . . 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 7437	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
11		prarloc2 7466	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
12	10, 11	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
13	8, 9, 12	syl2anc 409	. . . . . . 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 274	. . . . . . . . . 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 526	. . . . . . . . . 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 7443	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
17	10, 16	sylan 281	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
18	14, 15, 17	syl2anc 409	. . . . . . . . 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 524	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐴 ∈ P)
20	19	ad3antrrr 489	. . . . . . . . . 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 527	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
22	21	ad3antrrr 489	. . . . . . . . . 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 7443	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	1, 23	sylan 281	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	20, 22, 24	syl2anc 409	. . . . . . . . 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 7358	. . . . . . . . 9 ⊢ ((𝑢 ∈ Q ∧ 𝑥 ∈ Q) → (𝑢 <Q 𝑥 ∨ 𝑢 = 𝑥 ∨ 𝑥 <Q 𝑢))
27	18, 25, 26	syl2anc 409	. . . . . . . 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 274	. . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 7337	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
32	28, 30, 31	syl2anc 409	. . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
34	33	adantl 275	. . . . . . . . . . . . . . . . 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 7343	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
36	35	adantl 275	. . . . . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st ‘𝐴))
40	39	ad2antrr 485	. . . . . . . . . . . . . . . . . 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 2247	. . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . 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 149	. . . . . . . . . . . . . . 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 527	. . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . 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 123	. . . . . . . . . . . . 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 2233	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd ‘𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵)))
50		eleq1 2233	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st ‘𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
51	49, 50	anbi12d 470	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))))
52	51	rspcev 2834	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 7468	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴))))
55	14, 20, 54	syl2anc 409	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 166	. . . . . . . . . . 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 530	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
59	58	ad3antrrr 489	. . . . . . . . . . 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 615	. . . . . . . . . 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 114	. . . . . . . . 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 2233	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ (1st ‘𝐵) ↔ 𝑥 ∈ (1st ‘𝐵)))
63	15, 62	syl5ibcom 154	. . . . . . . . 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 7446	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
65	10, 64	sylan 281	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
66	14, 15, 65	syl2anc 409	. . . . . . . . 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 1299	. . . . . . . 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 2592	. . . . . 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 2592	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
71	4, 70	rexlimddv 2592	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐵))
72	71	expr 373	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
73	72	3impa 1189	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
74	73	ssrdv 3153	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 972   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ⊆ wss 3121  ⟨cop 3586   class class class wbr 3989  ‘cfv 5198  (class class class)co 5853  1^st c1st 6117  2^nd c2nd 6118  Qcnq 7242   +_Q cplq 7244   <_Q cltq 7247  Pcnp 7253  <_P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432

This theorem is referenced by:  aptipr  7603