Theorem ltexprlemru 7810

Description: Lemma for ltexpri 7811. 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 7703	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4771	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 114	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 7673	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnminu 7687	. . . . 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 7680	. . . . . . . . 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 7680	. . . . . . . . 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 7606	. . . . . . 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 7673	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21		prarloc 7701	. . . . . . . . 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 7704	. . . . . . . . . . . . . 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 7685	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
37	20, 36	syl3an1 1304	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
38	32, 34, 35, 37	syl3anc 1271	. . . . . . . . . . . 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 7685	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
44	5, 43	syl3an1 1304	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
45	32, 39, 42, 44	syl3anc 1271	. . . . . . . . . . . 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 7596	. . . . . . . . . . . . 13 ⊢ <Q Or Q
47		ltrelnq 7563	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
48	46, 47	sotri 5124	. . . . . . . . . . . 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 2653	. . . . . . . . . 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 7607	. . . . . . . . . 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 6014	. . . . . . . . . . . . 13 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
57	56	eqeq1d 2238	. . . . . . . . . . . 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 7679	. . . . . . . . . . . . . . . . 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 7579	. . . . . . . . . . . . 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 7580	. . . . . . . . . . . . 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 6192	. . . . . . . . . . 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 7683	. . . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . 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 2292	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
85		oveq1 6014	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
86	85	eleq1d 2298	. . . . . . . . . . . . . . . . 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 2891	. . . . . . . . . . . . . . 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 7797	. . . . . . . . . . . . 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 7806	. . . . . . . . . . . . . 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 7666	. . . . . . . . . . . . . 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 7573	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
100	98, 99	genppreclu 7713	. . . . . . . . . . . . 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 2307	. . . . . . . . . 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 2307	. . . . . . . . 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 2653	. . . . . . . 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 2656	. . . . . 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 2653	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
110	7, 109	rexlimddv 2653	. . 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 3230	1 ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  {crab 2512   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  2^nd c2nd 6291  Qcnq 7478   +_Q cplq 7480   <_Q cltq 7483  Pcnp 7489   +_P cpp 7491  <_P cltp 7493

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-iplp 7666  df-iltp 7668

This theorem is referenced by:  ltexpri  7811