Theorem aptiprleml 6737

Description: Lemma for aptipr 6739. (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 6573	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 6586	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
3	1, 2	sylan 267	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
4	3	ad2ant2rl 480	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ (1st ‘𝐴)𝑥 <Q 𝑠)
5		ltexnqi 6507	. . . . . . 7 ⊢ (𝑥 <Q 𝑠 → ∃𝑡 ∈ Q (𝑥 +Q 𝑡) = 𝑠)
6	5	ad2antll 460	. . . . . 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 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐵 ∈ P)
8	7	ad2antrr 457	. . . . . . . 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 483	. . . . . . . 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 6573	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
11		prarloc2 6602	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
12	10, 11	sylan 267	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐵)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐵))
13	8, 9, 12	syl2anc 391	. . . . . . 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 261	. . . . . . . . . 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 483	. . . . . . . . . 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 6579	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
17	10, 16	sylan 267	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → 𝑢 ∈ Q)
18	14, 15, 17	syl2anc 391	. . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝐴 ∈ P)
20	19	ad3antrrr 461	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
22	21	ad3antrrr 461	. . . . . . . . . 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 6579	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	1, 23	sylan 267	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	20, 22, 24	syl2anc 391	. . . . . . . . 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 6494	. . . . . . . . 9 ⊢ ((𝑢 ∈ Q ∧ 𝑥 ∈ Q) → (𝑢 <Q 𝑥 ∨ 𝑢 = 𝑥 ∨ 𝑥 <Q 𝑢))
27	18, 25, 26	syl2anc 391	. . . . . . . 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 261	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . 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 6473	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
32	28, 30, 31	syl2anc 391	. . . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
34	33	adantl 262	. . . . . . . . . . . . . . . . 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 6479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
36	35	adantl 262	. . . . . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . 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 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))) → (𝑥 +Q 𝑡) = 𝑠)
39		simprl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st ‘𝐴))
40	39	ad2antrr 457	. . . . . . . . . . . . . . . . . 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 2114	. . . . . . . . . . . . . . . . 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 6582	. . . . . . . . . . . . . . . . . 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 267	. . . . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . . . . 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 139	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 299	. . . . . . . . . . . . . 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 115	. . . . . . . . . . . . 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 2100	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd ‘𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐵)))
50		eleq1 2100	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st ‘𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴)))
51	49, 50	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st ‘𝐴))))
52	51	rspcev 2656	. . . . . . . . . . . . 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 391	. . . . . . . . . . . 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 6604	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ∃𝑣 ∈ Q (𝑣 ∈ (2nd ‘𝐵) ∧ 𝑣 ∈ (1st ‘𝐴))))
55	14, 20, 54	syl2anc 391	. . . . . . . . . . . . 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 261	. . . . . . . . . . . 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 156	. . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
59	58	ad3antrrr 461	. . . . . . . . . . 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 550	. . . . . . . . . 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 108	. . . . . . . . 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 2100	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ (1st ‘𝐵) ↔ 𝑥 ∈ (1st ‘𝐵)))
63	15, 62	syl5ibcom 144	. . . . . . . . 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 6582	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
65	10, 64	sylan 267	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝑢 ∈ (1st ‘𝐵)) → (𝑥 <Q 𝑢 → 𝑥 ∈ (1st ‘𝐵)))
66	14, 15, 65	syl2anc 391	. . . . . . . . 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 1199	. . . . . . . 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 2437	. . . . . 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 2437	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ (𝑠 ∈ (1st ‘𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st ‘𝐵))
71	4, 70	rexlimddv 2437	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (¬ 𝐵<P 𝐴 ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐵))
72	71	expr 357	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
73	72	3impa 1099	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st ‘𝐴) → 𝑥 ∈ (1st ‘𝐵)))
74	73	ssrdv 2951	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   +_Q cplq 6380   <_Q cltq 6383  Pcnp 6389  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iltp 6568

This theorem is referenced by:  aptipr  6739