Theorem ltexprlemru 7985

Description: Lemma for ltexpri 7986. 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 7878	. . . . . . . 8 ⊢ <P ⊆ (P × P)
2	1	brel 4829	. . . . . . 7 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simprd 114	. . . . . 6 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
4		prop 7848	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		prnminu 7862	. . . . 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 537	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9		elprnqu 7855	. . . . . . . . 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 513	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 ∈ Q)
12		elprnqu 7855	. . . . . . . . 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 7781	. . . . . . 7 ⊢ ((𝑡 ∈ Q ∧ 𝑤 ∈ Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣 ∈ Q (𝑡 +Q 𝑣) = 𝑤))
16	11, 14, 15	syl2anc 415	. . . . . 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 7848	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21		prarloc 7876	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑣 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑢 ∈ (2nd ‘𝐴)𝑢 <Q (𝑧 +Q 𝑣))
24	23	ad2ant2r 513	. . . . . 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 539	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7879	. . . . . . . . . . . . . 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 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)))) → 𝐴<P 𝐵)
33		simplrl 541	. . . . . . . . . . . . . 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 545	. . . . . . . . . . . . 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 7860	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
37	20, 36	syl3an1 1311	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) → 𝑧 <Q 𝑞)
38	32, 34, 35, 37	syl3anc 1278	. . . . . . . . . . . 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 546	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . 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 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)))) → 𝑡 ∈ (2nd ‘𝐵))
43		prltlu 7860	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
44	5, 43	syl3an1 1311	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ (1st ‘𝐵) ∧ 𝑡 ∈ (2nd ‘𝐵)) → 𝑞 <Q 𝑡)
45	32, 39, 42, 44	syl3anc 1278	. . . . . . . . . . . 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 7771	. . . . . . . . . . . . 13 ⊢ <Q Or Q
47		ltrelnq 7738	. . . . . . . . . . . . 13 ⊢ <Q ⊆ (Q × Q)
48	46, 47	sotri 5186	. . . . . . . . . . . 12 ⊢ ((𝑧 <Q 𝑞 ∧ 𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
49	38, 45, 48	syl2anc 415	. . . . . . . . . . 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 2673	. . . . . . . . . 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 7782	. . . . . . . . . 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 542	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 537	. . . . . . . . . . . 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 6095	. . . . . . . . . . . . 13 ⊢ ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
57	56	eqeq1d 2247	. . . . . . . . . . . 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 7854	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
63	62	ad2ant2r 513	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
64	63	adantlr 481	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 541	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 7754	. . . . . . . . . . . . 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 7755	. . . . . . . . . . . . 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 6273	. . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 7858	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . 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 2315	. . . . . . . . . . . . . 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 2301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
85		oveq1 6095	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
86	85	eleq1d 2307	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)))
87	84, 86	anbi12d 477	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵))))
88	87	spcegv 2913	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (1st ‘𝐴) → ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
89	88	anabsi5 585	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd ‘𝐵)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵)))
90	81, 83, 89	syl2anc 415	. . . . . . . . . . . . 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 7972	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd ‘𝐶) ↔ (𝑠 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd ‘𝐵))))
93	68, 90, 92	sylanbrc 421	. . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 7981	. . . . . . . . . . . . . 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 7841	. . . . . . . . . . . . . 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 7748	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +Q 𝑣) ∈ Q)
100	98, 99	genppreclu 7888	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 437	. . . . . . . . . . 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 2316	. . . . . . . . . 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 2316	. . . . . . . . 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 2673	. . . . . . . 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 2676	. . . . . 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 2673	. . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ (2nd ‘𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
110	7, 109	rexlimddv 2673	. . 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 3254	1 ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  {crab 2532   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  1^st c1st 6376  2^nd c2nd 6377  Qcnq 7653   +_Q cplq 7655   <_Q cltq 7658  Pcnp 7664   +_P cpp 7666  <_P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843

This theorem is used by:  ltexpri  7986