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Theorem cauappcvgprlem1 6630
 Description: Lemma for cauappcvgpr 6633. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (φ𝐹:QQ)
cauappcvgpr.app (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (φ𝑝 Q A <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
cauappcvgprlem.q (φ𝑄 Q)
cauappcvgprlem.r (φ𝑅 Q)
Assertion
Ref Expression
cauappcvgprlem1 (φ → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {u ∣ (𝐹𝑄) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩))
Distinct variable groups:   A,𝑝   𝐿,𝑝,𝑞   φ,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,u   𝑄,𝑝,𝑞,𝑙,u   𝑅,𝑝,𝑞,𝑙,u
Allowed substitution hints:   φ(u,𝑙)   A(u,𝑞,𝑙)   𝐿(u,𝑙)

Proof of Theorem cauappcvgprlem1
Dummy variables f g x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgprlem.r . . . . 5 (φ𝑅 Q)
2 halfnqq 6393 . . . . 5 (𝑅 Qx Q (x +Q x) = 𝑅)
31, 2syl 14 . . . 4 (φx Q (x +Q x) = 𝑅)
4 simprl 483 . . . . 5 ((φ (x Q (x +Q x) = 𝑅)) → x Q)
5 cauappcvgpr.app . . . . . . . . . . 11 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
65adantr 261 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → 𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
7 cauappcvgprlem.q . . . . . . . . . . . 12 (φ𝑄 Q)
87adantr 261 . . . . . . . . . . 11 ((φ (x Q (x +Q x) = 𝑅)) → 𝑄 Q)
9 fveq2 5121 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → (𝐹𝑝) = (𝐹𝑄))
10 oveq1 5462 . . . . . . . . . . . . . . 15 (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
1110oveq2d 5471 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑞) +Q (𝑄 +Q 𝑞)))
129, 11breq12d 3768 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞))))
139, 10oveq12d 5473 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))
1413breq2d 3767 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))))
1512, 14anbi12d 442 . . . . . . . . . . . 12 (𝑝 = 𝑄 → (((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) ↔ ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))))
16 fveq2 5121 . . . . . . . . . . . . . . 15 (𝑞 = x → (𝐹𝑞) = (𝐹x))
17 oveq2 5463 . . . . . . . . . . . . . . 15 (𝑞 = x → (𝑄 +Q 𝑞) = (𝑄 +Q x))
1816, 17oveq12d 5473 . . . . . . . . . . . . . 14 (𝑞 = x → ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹x) +Q (𝑄 +Q x)))
1918breq2d 3767 . . . . . . . . . . . . 13 (𝑞 = x → ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x))))
2017oveq2d 5471 . . . . . . . . . . . . . 14 (𝑞 = x → ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q x)))
2116, 20breq12d 3768 . . . . . . . . . . . . 13 (𝑞 = x → ((𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x))))
2219, 21anbi12d 442 . . . . . . . . . . . 12 (𝑞 = x → (((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))) ↔ ((𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)) (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x)))))
2315, 22rspc2v 2656 . . . . . . . . . . 11 ((𝑄 Q x Q) → (𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)) (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x)))))
248, 4, 23syl2anc 391 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → (𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)) (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x)))))
256, 24mpd 13 . . . . . . . . 9 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)) (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x))))
2625simpld 105 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)))
27 cauappcvgpr.f . . . . . . . . . . 11 (φ𝐹:QQ)
2827adantr 261 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → 𝐹:QQ)
2928, 4ffvelrnd 5246 . . . . . . . . 9 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹x) Q)
30 addassnqg 6366 . . . . . . . . 9 (((𝐹x) Q 𝑄 Q x Q) → (((𝐹x) +Q 𝑄) +Q x) = ((𝐹x) +Q (𝑄 +Q x)))
3129, 8, 4, 30syl3anc 1134 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q x) = ((𝐹x) +Q (𝑄 +Q x)))
3226, 31breqtrrd 3781 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) <Q (((𝐹x) +Q 𝑄) +Q x))
33 ltanqg 6384 . . . . . . . . 9 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
3433adantl 262 . . . . . . . 8 (((φ (x Q (x +Q x) = 𝑅)) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
3527, 7ffvelrnd 5246 . . . . . . . . 9 (φ → (𝐹𝑄) Q)
3635adantr 261 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) Q)
37 addclnq 6359 . . . . . . . . . 10 (((𝐹x) Q 𝑄 Q) → ((𝐹x) +Q 𝑄) Q)
3829, 8, 37syl2anc 391 . . . . . . . . 9 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹x) +Q 𝑄) Q)
39 addclnq 6359 . . . . . . . . 9 ((((𝐹x) +Q 𝑄) Q x Q) → (((𝐹x) +Q 𝑄) +Q x) Q)
4038, 4, 39syl2anc 391 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q x) Q)
41 addcomnqg 6365 . . . . . . . . 9 ((f Q g Q) → (f +Q g) = (g +Q f))
4241adantl 262 . . . . . . . 8 (((φ (x Q (x +Q x) = 𝑅)) (f Q g Q)) → (f +Q g) = (g +Q f))
4334, 36, 40, 4, 42caovord2d 5612 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) <Q (((𝐹x) +Q 𝑄) +Q x) ↔ ((𝐹𝑄) +Q x) <Q ((((𝐹x) +Q 𝑄) +Q x) +Q x)))
4432, 43mpbid 135 . . . . . 6 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) +Q x) <Q ((((𝐹x) +Q 𝑄) +Q x) +Q x))
45 addassnqg 6366 . . . . . . . 8 ((((𝐹x) +Q 𝑄) Q x Q x Q) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = (((𝐹x) +Q 𝑄) +Q (x +Q x)))
4638, 4, 4, 45syl3anc 1134 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = (((𝐹x) +Q 𝑄) +Q (x +Q x)))
47 simprr 484 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (x +Q x) = 𝑅)
4847oveq2d 5471 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q (x +Q x)) = (((𝐹x) +Q 𝑄) +Q 𝑅))
491adantr 261 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → 𝑅 Q)
50 addassnqg 6366 . . . . . . . 8 (((𝐹x) Q 𝑄 Q 𝑅 Q) → (((𝐹x) +Q 𝑄) +Q 𝑅) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5129, 8, 49, 50syl3anc 1134 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q 𝑅) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5246, 48, 513eqtrd 2073 . . . . . 6 ((φ (x Q (x +Q x) = 𝑅)) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5344, 52breqtrd 3779 . . . . 5 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅)))
54 oveq2 5463 . . . . . . 7 (𝑞 = x → ((𝐹𝑄) +Q 𝑞) = ((𝐹𝑄) +Q x))
5516oveq1d 5470 . . . . . . 7 (𝑞 = x → ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5654, 55breq12d 3768 . . . . . 6 (𝑞 = x → (((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅))))
5756rspcev 2650 . . . . 5 ((x Q ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅))) → 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
584, 53, 57syl2anc 391 . . . 4 ((φ (x Q (x +Q x) = 𝑅)) → 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
593, 58rexlimddv 2431 . . 3 (φ𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
60 cauappcvgpr.bnd . . . . . . . 8 (φ𝑝 Q A <Q (𝐹𝑝))
61 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
62 addclnq 6359 . . . . . . . . 9 ((𝑄 Q 𝑅 Q) → (𝑄 +Q 𝑅) Q)
637, 1, 62syl2anc 391 . . . . . . . 8 (φ → (𝑄 +Q 𝑅) Q)
6427, 5, 60, 61, 63cauappcvgprlemladd 6629 . . . . . . 7 (φ → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩)
6564fveq2d 5125 . . . . . 6 (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) = (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩))
66 nqex 6347 . . . . . . . 8 Q V
6766rabex 3892 . . . . . . 7 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} V
6866rabex 3892 . . . . . . 7 {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u} V
6967, 68op1st 5715 . . . . . 6 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}
7065, 69syl6eq 2085 . . . . 5 (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))})
7170eleq2d 2104 . . . 4 (φ → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ (𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}))
72 oveq1 5462 . . . . . . . 8 (𝑙 = (𝐹𝑄) → (𝑙 +Q 𝑞) = ((𝐹𝑄) +Q 𝑞))
7372breq1d 3765 . . . . . . 7 (𝑙 = (𝐹𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7473rexbidv 2321 . . . . . 6 (𝑙 = (𝐹𝑄) → (𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7574elrab3 2693 . . . . 5 ((𝐹𝑄) Q → ((𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7635, 75syl 14 . . . 4 (φ → ((𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7771, 76bitrd 177 . . 3 (φ → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7859, 77mpbird 156 . 2 (φ → (𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)))
7927, 5, 60, 61cauappcvgprlemcl 6624 . . . 4 (φ𝐿 P)
80 nqprlu 6529 . . . . 5 ((𝑄 +Q 𝑅) Q → ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩ P)
8163, 80syl 14 . . . 4 (φ → ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩ P)
82 addclpr 6519 . . . 4 ((𝐿 P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩ P) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) P)
8379, 81, 82syl2anc 391 . . 3 (φ → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) P)
84 nqprl 6532 . . 3 (((𝐹𝑄) Q (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) P) → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {u ∣ (𝐹𝑄) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)))
8535, 83, 84syl2anc 391 . 2 (φ → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {u ∣ (𝐹𝑄) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)))
8678, 85mpbid 135 1 (φ → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {u ∣ (𝐹𝑄) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264   +Q cplq 6266
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