Theorem ltexprlemru 7074

Description: Lemma for ltexpri 7075. 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 6967	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4448	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 112	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 6937	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnminu 6951	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
7	5, 6	sylan 277	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
8		simprr 499	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9		elprnqu 6944	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
10	5, 9	sylan 277	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
11	10	ad2ant2r 493	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 ∈ Q)
12		elprnqu 6944	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
13	5, 12	sylan 277	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
14	13	adantr 270	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ Q)
15		ltexnqq 6870	. . . . . . 7 ⊢ ((𝑡 ∈ Q ∧ 𝑤 ∈ Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤))
16	11, 14, 15	syl2anc 403	. . . . . 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 145	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤)
18	2	simpld 110	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
19		prop 6937	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21		prarloc 6965	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
22	20, 21	sylan 277	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
23	22	adantlr 461	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
24	23	ad2ant2r 493	. . . . . 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 500	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 6968	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
28	27	biimpd 142	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 → ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
29	2, 28	mpcom 36	. . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 472	. . . . . . . . . . . . 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 502	. . . . . . . . . . . . . 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 270	. . . . . . . . . . . . 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 506	. . . . . . . . . . . . 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 6949	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
37	20, 36	syl3an1 1203	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
38	32, 34, 35, 37	syl3anc 1170	. . . . . . . . . . . 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 507	. . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 472	. . . . . . . . . . . . 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 6949	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
44	5, 43	syl3an1 1203	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
45	32, 39, 42, 44	syl3anc 1170	. . . . . . . . . . . 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 6860	. . . . . . . . . . . . 13 ⊢ <Q Or Q
47		ltrelnq 6827	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
48	46, 47	sotri 4782	. . . . . . . . . . . 12 ⊢ ((𝑧 <Q 𝑞 ∧ 𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
49	38, 45, 48	syl2anc 403	. . . . . . . . . . 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 2487	. . . . . . . . . 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 6871	. . . . . . . . . 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 503	. . . . . . . . . . . 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 472	. . . . . . . . . . 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 499	. . . . . . . . . . . 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 5598	. . . . . . . . . . . . 13 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
57	56	eqeq1d 2091	. . . . . . . . . . . 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 165	. . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
61	20, 60	sylan 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
62	61	adantlr 461	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
63	62	ad2ant2r 493	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
64	63	adantlr 461	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 502	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 498	. . . . . . . . . . . 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 6843	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
70	69	adantl 271	. . . . . . . . . . . 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 6844	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
72	71	adantl 271	. . . . . . . . . . . 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 5760	. . . . . . . . . . 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 108	. . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . 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 6947	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
77	20, 76	sylan 277	. . . . . . . . . . . . . . 15 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
78	26, 75, 77	syl2anc 403	. . . . . . . . . . . . . 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 270	. . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . 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 2159	. . . . . . . . . . . . . 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 2145	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
85		oveq1 5598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
86	85	eleq1d 2151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)))
87	84, 86	anbi12d 457	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵))))
88	87	spcegv 2697	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (1st ‘𝐴) → ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
89	88	anabsi5 544	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)))
90	81, 83, 89	syl2anc 403	. . . . . . . . . . . . 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 7061	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
93	68, 90, 92	sylanbrc 408	. . . . . . . . . . . 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 472	. . . . . . . . . . . . . 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 7070	. . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . 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 6837	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
100	98, 99	genppreclu 6977	. . . . . . . . . . . . 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 403	. . . . . . . . . . . 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 424	. . . . . . . . . . 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 2160	. . . . . . . . . 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 2160	. . . . . . . . 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 2487	. . . . . . . 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 113	. . . . . . 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 2490	. . . . . 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 2487	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
110	7, 109	rexlimddv 2487	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111	110	ex 113	. 2 ⊢ (𝐴<P 𝐵 → (𝑤 ∈ (2nd ‘𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112	111	ssrdv 3016	1 ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 920   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ∃wrex 2354  {crab 2357   ⊆ wss 2984  ⟨cop 3425   class class class wbr 3811  ‘cfv 4969  (class class class)co 5591  1^st c1st 5844  2^nd c2nd 5845  Qcnq 6742   +_Q cplq 6744   <_Q cltq 6747  Pcnp 6753   +_P cpp 6755  <_P cltp 6757

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-iltp 6932

This theorem is referenced by:  ltexpri  7075