Theorem ltexprlemfu 7972

Description: Lemma for ltexpri 7974. 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 7866	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 4825	. . . . 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 7969	. . . 4 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7829	. . . . 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 7736	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
8	6, 7	genpelvu 7874	. . . 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 415	. . 3 ⊢ (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢)))
10		simprr 537	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
11	4	ltexprlemelu 7960	. . . . . . . . . . 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 493	. . . . . . . . 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 7836	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
17	3, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
18		prltlu 7848	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
19	17, 18	syl3an1 1311	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
20	19	3com23 1240	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
21	20	3adant2r 1264	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
22	21	3adant2r 1264	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
23	22	3adant3r 1266	. . . . . . . . 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 7761	. . . . . . . . . . . 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 7842	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑦 ∈ Q)
29	28	3adant2 1047	. . . . . . . . . . 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 7843	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑤 ∈ Q)
33	32	adantrr 483	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑤 ∈ Q)
34	33	3adant3 1048	. . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
37		elprnqu 7843	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑢 ∈ Q)
40	39	adantrr 483	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢 ∈ Q)
41	40	3adant3 1048	. . . . . . . . . . 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 7742	. . . . . . . . . . . 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 6253	. . . . . . . . . 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 7836	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
47	45, 46	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
48		prcunqu 7846	. . . . . . . . . . . . 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 482	. . . . . . . . . . 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 1047	. . . . . . . . . 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 1234	. . . . . . 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 1954	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
56	10, 55	eqeltrd 2315	. . . . 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 2676	. . 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 3254	1 ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  {crab 2532   ⊆ wss 3220  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   +_Q cplq 7643   <_Q cltq 7646  Pcnp 7652   +_P cpp 7654  <_P cltp 7656

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-iplp 7829  df-iltp 7831

This theorem is referenced by:  ltexpri  7974