Theorem ltexprlemfu 7671

Description: Lemma for ltexpri 7673. 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
ltexprlemfu	⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))

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

Proof of Theorem ltexprlemfu

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

Step	Hyp	Ref	Expression
1		ltrelpr 7565	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 4711	. . . . 5 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simpld 112	. . . 4 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
4		ltexprlem.1	. . . . 5 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
5	4	ltexprlempr 7668	. . . 4 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7528	. . . . 5 ⊢ +P = (𝑧 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑧) ∧ ℎ ∈ (1st ‘𝑦) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑧) ∧ ℎ ∈ (2nd ‘𝑦) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
7		addclnq 7435	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
8	6, 7	genpelvu 7573	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢)))
9	3, 5, 8	syl2anc 411	. . 3 ⊢ (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢)))
10		simprr 531	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
11	4	ltexprlemelu 7659	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (2nd ‘𝐶) ↔ (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))))
12	11	biimpi 120	. . . . . . . . . 10 ⊢ (𝑢 ∈ (2nd ‘𝐶) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))))
13	12	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))))
14	13	simprd 114	. . . . . . . 8 ⊢ (((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)))
15	14	adantl 277	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)))
16		prop 7535	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
17	3, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
18		prltlu 7547	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
19	17, 18	syl3an1 1282	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
20	19	3com23 1211	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
21	20	3adant2r 1235	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
22	21	3adant2r 1235	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
23	22	3adant3r 1237	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑦 <Q 𝑤)
24		ltanqg 7460	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
25	24	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
26		elprnql 7541	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
27	17, 26	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
28	27	adantrr 479	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑦 ∈ Q)
29	28	3adant2 1018	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑦 ∈ Q)
30		elprnqu 7542	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑤 ∈ Q)
31	17, 30	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑤 ∈ Q)
32	31	adantrr 479	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑤 ∈ Q)
33	32	adantrr 479	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑤 ∈ Q)
34	33	3adant3 1019	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑤 ∈ Q)
35		prop 7535	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
37		elprnqu 7542	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐶)) → 𝑢 ∈ Q)
38	36, 37	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐶)) → 𝑢 ∈ Q)
39	38	adantrl 478	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑢 ∈ Q)
40	39	adantrr 479	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢 ∈ Q)
41	40	3adant3 1019	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑢 ∈ Q)
42		addcomnqg 7441	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43	42	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
44	25, 29, 34, 41, 43	caovord2d 6088	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → (𝑦 <Q 𝑤 ↔ (𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢)))
45	2	simprd 114	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
46		prop 7535	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
47	45, 46	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
48		prcunqu 7545	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
49	47, 48	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
50	49	adantrl 478	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
51	50	3adant2 1018	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
52	44, 51	sylbid 150	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → (𝑦 <Q 𝑤 → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
53	23, 52	mpd 13	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
54	53	3expa 1205	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
55	15, 54	exlimddv 1910	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
56	10, 55	eqeltrd 2270	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (2nd ‘𝐵))
57	56	expr 375	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd ‘𝐵)))
58	57	rexlimdvva 2619	. . 3 ⊢ (𝐴<P 𝐵 → (∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd ‘𝐵)))
59	9, 58	sylbid 150	. 2 ⊢ (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (2nd ‘𝐵)))
60	59	ssrdv 3185	1 ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473  {crab 2476   ⊆ wss 3153  ⟨cop 3621   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340   +_Q cplq 7342   <_Q cltq 7345  Pcnp 7351   +_P cpp 7353  <_P cltp 7355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  ltexpri  7673