Theorem ltexprlemru 6710

Description: Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩

Assertion

Ref	Expression
ltexprlemru	⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemru

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

Step	Hyp	Ref	Expression
1		ltrelpr 6603	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4392	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 107	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 6573	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnminu 6587	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
7	5, 6	sylan 267	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
8		simprr 484	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9		elprnqu 6580	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
10	5, 9	sylan 267	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
11	10	ad2ant2r 478	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 ∈ Q)
12		elprnqu 6580	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
13	5, 12	sylan 267	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
14	13	adantr 261	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ Q)
15		ltexnqq 6506	. . . . . . 7 ⊢ ((𝑡 ∈ Q ∧ 𝑤 ∈ Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤))
16	11, 14, 15	syl2anc 391	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → (𝑡 <Q 𝑤 ↔ ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤))
17	8, 16	mpbid 135	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤)
18	2	simpld 105	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
19		prop 6573	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21		prarloc 6601	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
22	20, 21	sylan 267	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
23	22	adantlr 446	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
24	23	ad2ant2r 478	. . . . . 6 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
25		simplll 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝐴<P 𝐵)
26	25	ad2antrr 457	. . . . . . . . . . . 12 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
27		ltdfpr 6604	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
28	27	biimpd 132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
29	2, 28	mpcom 32	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))
30	26, 29	syl 14	. . . . . . . . . . 11 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))
31	25	adantr 261	. . . . . . . . . . . . . 14 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝐴<P 𝐵)
32	31	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝐴<P 𝐵)
33		simplrl 487	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st ‘𝐴))
34	33	adantr 261	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑧 ∈ (1st ‘𝐴))
35		simprrl 491	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑞 ∈ (2nd ‘𝐴))
36		prltlu 6585	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
37	20, 36	syl3an1 1168	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
38	32, 34, 35, 37	syl3anc 1135	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑧 <Q 𝑞)
39		simprrr 492	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑞 ∈ (1st ‘𝐵))
40		simplrl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑡 ∈ (2nd ‘𝐵))
41	40	adantr 261	. . . . . . . . . . . . . 14 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑡 ∈ (2nd ‘𝐵))
42	41	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑡 ∈ (2nd ‘𝐵))
43		prltlu 6585	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
44	5, 43	syl3an1 1168	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
45	32, 39, 42, 44	syl3anc 1135	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑞 <Q 𝑡)
46		ltsonq 6496	. . . . . . . . . . . . 13 ⊢ <Q Or Q
47		ltrelnq 6463	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
48	46, 47	sotri 4720	. . . . . . . . . . . 12 ⊢ ((𝑧 <Q 𝑞 ∧ 𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
49	38, 45, 48	syl2anc 391	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞 ∈ Q ∧ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) → 𝑧 <Q 𝑡)
50	30, 49	rexlimddv 2437	. . . . . . . . . 10 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 <Q 𝑡)
51		ltexnqi 6507	. . . . . . . . . 10 ⊢ (𝑧 <Q 𝑡 → ∃𝑠 ∈ Q (𝑧 +Q 𝑠) = 𝑡)
52	50, 51	syl 14	. . . . . . . . 9 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ∃𝑠 ∈ Q (𝑧 +Q 𝑠) = 𝑡)
53		simplrr 488	. . . . . . . . . . . 12 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → (𝑡 +Q 𝑣) = 𝑤)
54	53	ad2antrr 457	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑡 +Q 𝑣) = 𝑤)
55		simprr 484	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑠) = 𝑡)
56		oveq1 5519	. . . . . . . . . . . . 13 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
57	56	eqeq1d 2048	. . . . . . . . . . . 12 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → (((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤 ↔ (𝑡 +Q 𝑣) = 𝑤))
58	55, 57	syl 14	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤 ↔ (𝑡 +Q 𝑣) = 𝑤))
59	54, 58	mpbird 156	. . . . . . . . . 10 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤)
60		elprnql 6579	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
61	20, 60	sylan 267	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
62	61	adantlr 446	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
63	62	ad2ant2r 478	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
64	63	adantlr 446	. . . . . . . . . . . . 13 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
65	64	ad2antrr 457	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑧 ∈ Q)
66		simplrl 487	. . . . . . . . . . . . 13 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
67	66	ad2antrr 457	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑣 ∈ Q)
68		simprl 483	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑠 ∈ Q)
69		addcomnqg 6479	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
70	69	adantl 262	. . . . . . . . . . . 12 ⊢ ((((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
71		addassnqg 6480	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
72	71	adantl 262	. . . . . . . . . . . 12 ⊢ ((((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
73	65, 67, 68, 70, 72	caov32d 5681	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑣) +Q 𝑠) = ((𝑧 +Q 𝑠) +Q 𝑣))
74		simpr 103	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 <Q (𝑧 +Q 𝑣))
75		simplrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd ‘𝐴))
76		prcunqu 6583	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
77	20, 76	sylan 267	. . . . . . . . . . . . . . 15 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
78	26, 75, 77	syl2anc 391	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
79	74, 78	mpd 13	. . . . . . . . . . . . 13 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴))
80	79	adantr 261	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴))
81	33	adantr 261	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑧 ∈ (1st ‘𝐴))
82	41	ad2antrr 457	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑡 ∈ (2nd ‘𝐵))
83	55, 82	eqeltrd 2114	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵))
84		eleq1 2100	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
85		oveq1 5519	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
86	85	eleq1d 2106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)))
87	84, 86	anbi12d 442	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵))))
88	87	spcegv 2641	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (1st ‘𝐴) → ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
89	88	anabsi5 513	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)))
90	81, 83, 89	syl2anc 391	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)))
91		ltexprlem.1	. . . . . . . . . . . . . 14 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
92	91	ltexprlemelu 6697	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
93	68, 90, 92	sylanbrc 394	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑠 ∈ (2nd ‘𝐶))
94	31	ad2antrr 457	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐴<P 𝐵)
95	94, 18	syl 14	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐴 ∈ P)
96	91	ltexprlempr 6706	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
97	94, 96	syl 14	. . . . . . . . . . . . 13 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐶 ∈ P)
98		df-iplp 6566	. . . . . . . . . . . . . 14 ⊢ +P = (𝑥 ∈ P, 𝑤 ∈ P ↦ ⟨{𝑧 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑣 ∈ Q (𝑓 ∈ (1st ‘𝑥) ∧ 𝑣 ∈ (1st ‘𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}, {𝑧 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑣 ∈ Q (𝑓 ∈ (2nd ‘𝑥) ∧ 𝑣 ∈ (2nd ‘𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}⟩)
99		addclnq 6473	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
100	98, 99	genppreclu 6613	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) ∧ 𝑠 ∈ (2nd ‘𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
101	95, 97, 100	syl2anc 391	. . . . . . . . . . . 12 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (((𝑧 +Q 𝑣) ∈ (2nd ‘𝐴) ∧ 𝑠 ∈ (2nd ‘𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
102	80, 93, 101	mp2and 409	. . . . . . . . . . 11 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶)))
103	73, 102	eqeltrrd 2115	. . . . . . . . . 10 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑠) +Q 𝑣) ∈ (2nd ‘(𝐴 +P 𝐶)))
104	59, 103	eqeltrrd 2115	. . . . . . . . 9 ⊢ (((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
105	52, 104	rexlimddv 2437	. . . . . . . 8 ⊢ ((((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
106	105	ex 108	. . . . . . 7 ⊢ (((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → (𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
107	106	rexlimdvva 2440	. . . . . 6 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → (∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
108	24, 107	mpd 13	. . . . 5 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣 ∈ Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
109	17, 108	rexlimddv 2437	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
110	7, 109	rexlimddv 2437	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111	110	ex 108	. 2 ⊢ (𝐴<P 𝐵 → (𝑤 ∈ (2nd ‘𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112	111	ssrdv 2951	1 ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  {crab 2310   ⊆ 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 cpp 6391  <_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-iplp 6566  df-iltp 6568

This theorem is referenced by:  ltexpri  6711