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Theorem addnqprlemru 7588
Description: Lemma for addnqpr 7591. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemru ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem addnqprlemru
Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 7577 . . . . . 6 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
2 nqprlu 7577 . . . . . 6 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
3 df-iplp 7498 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
4 addclnq 7405 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
53, 4genpelvu 7543 . . . . . 6 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
61, 2, 5syl2an 289 . . . . 5 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
76biimpa 296 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
8 vex 2755 . . . . . . . . . . . . 13 𝑠 ∈ V
9 breq2 4022 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → (𝐴 <Q 𝑢𝐴 <Q 𝑠))
10 ltnqex 7579 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐴} ∈ V
11 gtnqex 7580 . . . . . . . . . . . . . 14 {𝑢𝐴 <Q 𝑢} ∈ V
1210, 11op2nd 6173 . . . . . . . . . . . . 13 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
138, 9, 12elab2 2900 . . . . . . . . . . . 12 (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑠)
1413biimpi 120 . . . . . . . . . . 11 (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑠)
1514ad2antrl 490 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝐴 <Q 𝑠)
1615adantr 276 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝐴 <Q 𝑠)
17 vex 2755 . . . . . . . . . . . . 13 𝑡 ∈ V
18 breq2 4022 . . . . . . . . . . . . 13 (𝑢 = 𝑡 → (𝐵 <Q 𝑢𝐵 <Q 𝑡))
19 ltnqex 7579 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐵} ∈ V
20 gtnqex 7580 . . . . . . . . . . . . . 14 {𝑢𝐵 <Q 𝑢} ∈ V
2119, 20op2nd 6173 . . . . . . . . . . . . 13 (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑢𝐵 <Q 𝑢}
2217, 18, 21elab2 2900 . . . . . . . . . . . 12 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝐵 <Q 𝑡)
2322biimpi 120 . . . . . . . . . . 11 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) → 𝐵 <Q 𝑡)
2423ad2antll 491 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝐵 <Q 𝑡)
2524adantr 276 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝐵 <Q 𝑡)
26 ltrelnq 7395 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4696 . . . . . . . . . . 11 (𝐴 <Q 𝑠 → (𝐴Q𝑠Q))
2816, 27syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴Q𝑠Q))
2926brel 4696 . . . . . . . . . . 11 (𝐵 <Q 𝑡 → (𝐵Q𝑡Q))
3025, 29syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐵Q𝑡Q))
31 lt2addnq 7434 . . . . . . . . . 10 (((𝐴Q𝑠Q) ∧ (𝐵Q𝑡Q)) → ((𝐴 <Q 𝑠𝐵 <Q 𝑡) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
3228, 30, 31syl2anc 411 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝐴 <Q 𝑠𝐵 <Q 𝑡) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
3316, 25, 32mp2and 433 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡))
34 breq2 4022 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → ((𝐴 +Q 𝐵) <Q 𝑟 ↔ (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
3534adantl 277 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝐴 +Q 𝐵) <Q 𝑟 ↔ (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
3633, 35mpbird 167 . . . . . . 7 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴 +Q 𝐵) <Q 𝑟)
37 vex 2755 . . . . . . . 8 𝑟 ∈ V
38 breq2 4022 . . . . . . . 8 (𝑢 = 𝑟 → ((𝐴 +Q 𝐵) <Q 𝑢 ↔ (𝐴 +Q 𝐵) <Q 𝑟))
39 ltnqex 7579 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
40 gtnqex 7580 . . . . . . . . 9 {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
4139, 40op2nd 6173 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}
4237, 38, 41elab2 2900 . . . . . . 7 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) <Q 𝑟)
4336, 42sylibr 134 . . . . . 6 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
4443ex 115 . . . . 5 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
4544rexlimdvva 2615 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
467, 45mpd 13 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
4746ex 115 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
4847ssrdv 3176 1 ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  {cab 2175  wrex 2469  wss 3144  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5897  2nd c2nd 6165  Qcnq 7310   +Q cplq 7312   <Q cltq 7315  Pcnp 7321   +P cpp 7323
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-iplp 7498
This theorem is referenced by:  addnqprlemfl  7589  addnqpr  7591
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