Theorem aptiprleml 7471

Description: Lemma for aptipr 7473. (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 7307	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7320	. . . . . . 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 503	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
5		ltexnqi 7241	. . . . . . 7 ⊢ (𝑥 <Q 𝑠 → ∃𝑡 ∈ Q (𝑥 +Q 𝑡) = 𝑠)
6	5	ad2antll 483	. . . . . 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 520	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐵 ∈ P)
8	7	ad2antrr 480	. . . . . . . 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 521	. . . . . . . 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 7307	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
11		prarloc2 7336	. . . . . . . . 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 521	. . . . . . . . . 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 7313	. . . . . . . . . . 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 519	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐴 ∈ P)
20	19	ad3antrrr 484	. . . . . . . . . 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 522	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
22	21	ad3antrrr 484	. . . . . . . . . 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 7313	. . . . . . . . . . 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 7228	. . . . . . . . 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 525	. . . . . . . . . . . . . . 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 7207	. . . . . . . . . . . . . 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 7232	. . . . . . . . . . . . . . . . . 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 7213	. . . . . . . . . . . . . . . . . 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 5948	. . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st ‘𝐴))
40	39	ad2antrr 480	. . . . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . . . 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 7316	. . . . . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . . 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 2203	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd ‘𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵)))
50		eleq1 2203	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st ‘𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
51	49, 50	anbi12d 465	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))))
52	51	rspcev 2793	. . . . . . . . . . . . 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 7338	. . . . . . . . . . . . . 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 525	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
59	58	ad3antrrr 484	. . . . . . . . . . 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 610	. . . . . . . . . 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 2203	. . . . . . . . . 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 7316	. . . . . . . . . . 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 1283	. . . . . . . 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 2557	. . . . . 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 2557	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
71	4, 70	rexlimddv 2557	. . . 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 1177	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
74	73	ssrdv 3108	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 962   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ⊆ wss 3076  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  1^st c1st 6044  2^nd c2nd 6045  Qcnq 7112   +_Q cplq 7114   <_Q cltq 7117  Pcnp 7123  <_P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iltp 7302

This theorem is referenced by:  aptipr  7473