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Theorem addnqprlemrl 7019
Description: Lemma for addnqpr 7023. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemrl ((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem addnqprlemrl
Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 7009 . . . . . 6 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
2 nqprlu 7009 . . . . . 6 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
3 df-iplp 6930 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
4 addclnq 6837 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
53, 4genpelvl 6974 . . . . . 6 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
61, 2, 5syl2an 283 . . . . 5 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
76biimpa 290 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
8 vex 2615 . . . . . . . . . . . . 13 𝑠 ∈ V
9 breq1 3814 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 <Q 𝐴𝑠 <Q 𝐴))
10 ltnqex 7011 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐴} ∈ V
11 gtnqex 7012 . . . . . . . . . . . . . 14 {𝑢𝐴 <Q 𝑢} ∈ V
1210, 11op1st 5852 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
138, 9, 12elab2 2751 . . . . . . . . . . . 12 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑠 <Q 𝐴)
1413biimpi 118 . . . . . . . . . . 11 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝑠 <Q 𝐴)
1514ad2antrl 474 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑠 <Q 𝐴)
1615adantr 270 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 <Q 𝐴)
17 vex 2615 . . . . . . . . . . . . 13 𝑡 ∈ V
18 breq1 3814 . . . . . . . . . . . . 13 (𝑙 = 𝑡 → (𝑙 <Q 𝐵𝑡 <Q 𝐵))
19 ltnqex 7011 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐵} ∈ V
20 gtnqex 7012 . . . . . . . . . . . . . 14 {𝑢𝐵 <Q 𝑢} ∈ V
2119, 20op1st 5852 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐵}
2217, 18, 21elab2 2751 . . . . . . . . . . . 12 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝐵)
2322biimpi 118 . . . . . . . . . . 11 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) → 𝑡 <Q 𝐵)
2423ad2antll 475 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑡 <Q 𝐵)
2524adantr 270 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 <Q 𝐵)
26 ltrelnq 6827 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4448 . . . . . . . . . . 11 (𝑠 <Q 𝐴 → (𝑠Q𝐴Q))
2816, 27syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠Q𝐴Q))
2926brel 4448 . . . . . . . . . . 11 (𝑡 <Q 𝐵 → (𝑡Q𝐵Q))
3025, 29syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑡Q𝐵Q))
31 lt2addnq 6866 . . . . . . . . . 10 (((𝑠Q𝐴Q) ∧ (𝑡Q𝐵Q)) → ((𝑠 <Q 𝐴𝑡 <Q 𝐵) → (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3228, 30, 31syl2anc 403 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝑠 <Q 𝐴𝑡 <Q 𝐵) → (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3316, 25, 32mp2and 424 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵))
34 breq1 3814 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (𝐴 +Q 𝐵) ↔ (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3534adantl 271 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 <Q (𝐴 +Q 𝐵) ↔ (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3633, 35mpbird 165 . . . . . . 7 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 <Q (𝐴 +Q 𝐵))
37 vex 2615 . . . . . . . 8 𝑟 ∈ V
38 breq1 3814 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 <Q (𝐴 +Q 𝐵) ↔ 𝑟 <Q (𝐴 +Q 𝐵)))
39 ltnqex 7011 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
40 gtnqex 7012 . . . . . . . . 9 {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
4139, 40op1st 5852 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 +Q 𝐵)}
4237, 38, 41elab2 2751 . . . . . . 7 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 +Q 𝐵))
4336, 42sylibr 132 . . . . . 6 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
4443ex 113 . . . . 5 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
4544rexlimdvva 2490 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
467, 45mpd 13 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
4746ex 113 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
4847ssrdv 3016 1 ((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  {cab 2069  wrex 2354  wss 2984  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  Qcnq 6742   +Q cplq 6744   <Q cltq 6747  Pcnp 6753   +P cpp 6755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-inp 6928  df-iplp 6930
This theorem is referenced by:  addnqprlemfu  7022  addnqpr  7023
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