Theorem ltexprlemfu 7931

Description: Lemma for ltexpri 7933. 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 7825	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 4804	. . . . 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 7928	. . . 4 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
6		df-iplp 7788	. . . . 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 7695	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
8	6, 7	genpelvu 7833	. . . 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 533	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
11	4	ltexprlemelu 7919	. . . . . . . . . . 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 7795	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
17	3, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
18		prltlu 7807	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
19	17, 18	syl3an1 1307	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
20	19	3com23 1236	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
21	20	3adant2r 1260	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
22	21	3adant2r 1260	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
23	22	3adant3r 1262	. . . . . . . . 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 7720	. . . . . . . . . . . 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 7801	. . . . . . . . . . . . . 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 1043	. . . . . . . . . . 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 7802	. . . . . . . . . . . . . . 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 1044	. . . . . . . . . . 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 7795	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
37		elprnqu 7802	. . . . . . . . . . . . . . 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 1044	. . . . . . . . . . 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 7701	. . . . . . . . . . . 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 6226	. . . . . . . . . 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 7795	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
47	45, 46	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
48		prcunqu 7805	. . . . . . . . . . . . 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 1043	. . . . . . . . . 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 1230	. . . . . . 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 1950	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
56	10, 55	eqeltrd 2311	. . . . 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 2670	. . 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 3246	1 ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523  {crab 2526   ⊆ wss 3213  ⟨cop 3694   class class class wbr 4111  ‘cfv 5354  (class class class)co 6052  1^st c1st 6334  2^nd c2nd 6335  Qcnq 7600   +_Q cplq 7602   <_Q cltq 7605  Pcnp 7611   +_P cpp 7613  <_P cltp 7615

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673  df-enq0 7744  df-nq0 7745  df-0nq0 7746  df-plq0 7747  df-mq0 7748  df-inp 7786  df-iplp 7788  df-iltp 7790

This theorem is referenced by:  ltexpri  7933