Theorem ltexprlemru 7679

Description: Lemma for ltexpri 7680. 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 7572	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4715	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 114	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 7542	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnminu 7556	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
7	5, 6	sylan 283	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → ∃𝑡 ∈ (2nd ‘𝐵)𝑡 <Q 𝑤)
8		simprr 531	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9		elprnqu 7549	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
10	5, 9	sylan 283	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑡 ∈ Q)
11	10	ad2ant2r 509	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 ∈ Q)
12		elprnqu 7549	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
13	5, 12	sylan 283	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ Q)
14	13	adantr 276	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ Q)
15		ltexnqq 7475	. . . . . . 7 ⊢ ((𝑡 ∈ Q ∧ 𝑤 ∈ Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤))
16	11, 14, 15	syl2anc 411	. . . . . 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 147	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤)
18	2	simpld 112	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
19		prop 7542	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21		prarloc 7570	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
22	20, 21	sylan 283	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
23	22	adantlr 477	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
24	23	ad2ant2r 509	. . . . . 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 533	. . . . . . . . . . . . 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 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))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
27		ltdfpr 7573	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
28	27	biimpd 144	. . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 535	. . . . . . . . . . . . . 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 276	. . . . . . . . . . . . 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 539	. . . . . . . . . . . . 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 7554	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
37	20, 36	syl3an1 1282	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
38	32, 34, 35, 37	syl3anc 1249	. . . . . . . . . . . 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 540	. . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 7554	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
44	5, 43	syl3an1 1282	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
45	32, 39, 42, 44	syl3anc 1249	. . . . . . . . . . . 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 7465	. . . . . . . . . . . . 13 ⊢ <Q Or Q
47		ltrelnq 7432	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
48	46, 47	sotri 5065	. . . . . . . . . . . 12 ⊢ ((𝑧 <Q 𝑞 ∧ 𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
49	38, 45, 48	syl2anc 411	. . . . . . . . . . 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 2619	. . . . . . . . . 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 7476	. . . . . . . . . 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 536	. . . . . . . . . . . 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 488	. . . . . . . . . . 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 531	. . . . . . . . . . . 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 5929	. . . . . . . . . . . . 13 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
57	56	eqeq1d 2205	. . . . . . . . . . . 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 167	. . . . . . . . . 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 7548	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
61	20, 60	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
62	61	adantlr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
63	62	ad2ant2r 509	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
64	63	adantlr 477	. . . . . . . . . . . . 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 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))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑧 ∈ Q)
66		simplrl 535	. . . . . . . . . . . . 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 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))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑣 ∈ Q)
68		simprl 529	. . . . . . . . . . . 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 7448	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
70	69	adantl 277	. . . . . . . . . . . 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 7449	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
72	71	adantl 277	. . . . . . . . . . . 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 6104	. . . . . . . . . . 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 110	. . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 7552	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
77	20, 76	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd ‘𝐴)))
78	26, 75, 77	syl2anc 411	. . . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 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)) ∧ (𝑠 ∈ Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑡 ∈ (2nd ‘𝐵))
83	55, 82	eqeltrd 2273	. . . . . . . . . . . . . 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 2259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
85		oveq1 5929	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
86	85	eleq1d 2265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)))
87	84, 86	anbi12d 473	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵))))
88	87	spcegv 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (1st ‘𝐴) → ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
89	88	anabsi5 579	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)))
90	81, 83, 89	syl2anc 411	. . . . . . . . . . . . 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 7666	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
93	68, 90, 92	sylanbrc 417	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 7675	. . . . . . . . . . . . . 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 7535	. . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
100	98, 99	genppreclu 7582	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 433	. . . . . . . . . . 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 2274	. . . . . . . . . 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 2274	. . . . . . . . 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 2619	. . . . . . . 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 115	. . . . . . 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 2622	. . . . . 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 2619	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
110	7, 109	rexlimddv 2619	. . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111	110	ex 115	. 2 ⊢ (𝐴<P 𝐵 → (𝑤 ∈ (2nd ‘𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112	111	ssrdv 3189	1 ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  {crab 2479   ⊆ wss 3157  ⟨cop 3625   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  1^st c1st 6196  2^nd c2nd 6197  Qcnq 7347   +_Q cplq 7349   <_Q cltq 7352  Pcnp 7358   +_P cpp 7360  <_P cltp 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-iplp 7535  df-iltp 7537

This theorem is referenced by:  ltexpri  7680