Theorem ltexprlemfu 6709

Description: Lemma for ltexpri 6711. 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 6603	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 4392	. . . . 5 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3	2	simpld 105	. . . 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 6706	. . . 4 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
6		df-iplp 6566	. . . . 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 6473	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
8	6, 7	genpelvu 6611	. . . 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 391	. . 3 ⊢ (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢)))
10		simprr 484	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
11	4	ltexprlemelu 6697	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (2nd ‘𝐶) ↔ (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))))
12	11	biimpi 113	. . . . . . . . . 10 ⊢ (𝑢 ∈ (2nd ‘𝐶) → (𝑢 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))))
13	12	ad2antlr 458	. . . . . . . . 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 107	. . . . . . . 8 ⊢ (((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)))
15	14	adantl 262	. . . . . . 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 6573	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
17	3, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
18		prltlu 6585	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
19	17, 18	syl3an1 1168	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑤)
20	19	3com23 1110	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
21	20	3adant2r 1130	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
22	21	3adant2r 1130	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 <Q 𝑤)
23	22	3adant3r 1132	. . . . . . . . 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 6498	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
25	24	adantl 262	. . . . . . . . . . 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 6579	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
27	17, 26	sylan 267	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
28	27	adantrr 448	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵))) → 𝑦 ∈ Q)
29	28	3adant2 923	. . . . . . . . . . 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 6580	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑤 ∈ Q)
31	17, 30	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ (2nd ‘𝐴)) → 𝑤 ∈ Q)
32	31	adantrr 448	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑤 ∈ Q)
33	32	adantrr 448	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑤 ∈ Q)
34	33	3adant3 924	. . . . . . . . . . 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 6573	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
37		elprnqu 6580	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑢 ∈ (2nd ‘𝐶)) → 𝑢 ∈ Q)
38	36, 37	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑢 ∈ (2nd ‘𝐶)) → 𝑢 ∈ Q)
39	38	adantrl 447	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → 𝑢 ∈ Q)
40	39	adantrr 448	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢 ∈ Q)
41	40	3adant3 924	. . . . . . . . . . 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 6479	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43	42	adantl 262	. . . . . . . . . . 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 5670	. . . . . . . . . 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 107	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
46		prop 6573	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
47	45, 46	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
48		prcunqu 6583	. . . . . . . . . . . . 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 267	. . . . . . . . . . . 12 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (2nd ‘𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵)))
50	49	adantrl 447	. . . . . . . . . . 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 923	. . . . . . . . . 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 139	. . . . . . . . 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 1104	. . . . . . 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 1778	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd ‘𝐵))
56	10, 55	eqeltrd 2114	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (2nd ‘𝐵))
57	56	expr 357	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd ‘𝐵)))
58	57	rexlimdvva 2440	. . 3 ⊢ (𝐴<P 𝐵 → (∃𝑤 ∈ (2nd ‘𝐴)∃𝑢 ∈ (2nd ‘𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd ‘𝐵)))
59	9, 58	sylbid 139	. 2 ⊢ (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (2nd ‘𝐵)))
60	59	ssrdv 2951	1 ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  {crab 2310   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   +_Q cplq 6380   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568

This theorem is referenced by:  ltexpri  6711