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Theorem prplnqu 6872
Description: Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
Hypotheses
Ref Expression
prplnqu.x (𝜑𝑋P)
prplnqu.q (𝜑𝑄Q)
prplnqu.sum (𝜑𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
Assertion
Ref Expression
prplnqu (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
Distinct variable groups:   𝐴,𝑙,𝑢   𝑦,𝐴   𝑄,𝑙,𝑢   𝑦,𝑄   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑙)   𝑋(𝑢,𝑙)

Proof of Theorem prplnqu
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prplnqu.q . . . . . . . 8 (𝜑𝑄Q)
2 nqprlu 6799 . . . . . . . 8 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
31, 2syl 14 . . . . . . 7 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
4 prplnqu.x . . . . . . 7 (𝜑𝑋P)
5 ltaddpr 6849 . . . . . . 7 ((⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P𝑋P) → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋))
63, 4, 5syl2anc 403 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋))
7 addcomprg 6830 . . . . . . 7 ((⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P𝑋P) → (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
83, 4, 7syl2anc 403 . . . . . 6 (𝜑 → (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
96, 8breqtrd 3817 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
10 prplnqu.sum . . . . . 6 (𝜑𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
11 addclpr 6789 . . . . . . . . 9 ((𝑋P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
124, 3, 11syl2anc 403 . . . . . . . 8 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
13 prop 6727 . . . . . . . . 9 ((𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)), (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))⟩ ∈ P)
14 elprnqu 6734 . . . . . . . . 9 ((⟨(1st ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)), (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))⟩ ∈ P𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))) → 𝐴Q)
1513, 14sylan 277 . . . . . . . 8 (((𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))) → 𝐴Q)
1612, 10, 15syl2anc 403 . . . . . . 7 (𝜑𝐴Q)
17 nqpru 6804 . . . . . . 7 ((𝐴Q ∧ (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P) → (𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)) ↔ (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1816, 12, 17syl2anc 403 . . . . . 6 (𝜑 → (𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)) ↔ (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1910, 18mpbid 145 . . . . 5 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
20 ltsopr 6848 . . . . . 6 <P Or P
21 ltrelpr 6757 . . . . . 6 <P ⊆ (P × P)
2220, 21sotri 4750 . . . . 5 ((⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∧ (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
239, 19, 22syl2anc 403 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
24 ltnqpr 6845 . . . . 5 ((𝑄Q𝐴Q) → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
251, 16, 24syl2anc 403 . . . 4 (𝜑 → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
2623, 25mpbird 165 . . 3 (𝜑𝑄 <Q 𝐴)
27 ltexnqi 6661 . . 3 (𝑄 <Q 𝐴 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2826, 27syl 14 . 2 (𝜑 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2919adantr 270 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
301adantr 270 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄Q)
31 simprl 498 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧Q)
32 addcomnqg 6633 . . . . . . . . . 10 ((𝑄Q𝑧Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
3330, 31, 32syl2anc 403 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34 simprr 499 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
3533, 34eqtr3d 2116 . . . . . . . 8 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36 breq2 3797 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
3736abbidv 2197 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑙𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙𝑙 <Q 𝐴})
38 breq1 3796 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢𝐴 <Q 𝑢))
3938abbidv 2197 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢𝐴 <Q 𝑢})
4037, 39opeq12d 3586 . . . . . . . 8 ((𝑧 +Q 𝑄) = 𝐴 → ⟨{𝑙𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
4135, 40syl 14 . . . . . . 7 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
42 addnqpr 6813 . . . . . . . 8 ((𝑧Q𝑄Q) → ⟨{𝑙𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
4331, 30, 42syl2anc 403 . . . . . . 7 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
4441, 43eqtr3d 2116 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
4529, 44breqtrd 3817 . . . . 5 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P (⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
46 ltaprg 6871 . . . . . . 7 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
4746adantl 271 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
484adantr 270 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋P)
49 nqprlu 6799 . . . . . . 7 (𝑧Q → ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ ∈ P)
5031, 49syl 14 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ ∈ P)
5130, 2syl 14 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
52 addcomprg 6830 . . . . . . 7 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5352adantl 271 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5447, 48, 50, 51, 53caovord2d 5701 . . . . 5 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ ↔ (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P (⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
5545, 54mpbird 165 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩)
56 nqpru 6804 . . . . 5 ((𝑧Q𝑋P) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5731, 48, 56syl2anc 403 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5855, 57mpbird 165 . . 3 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd𝑋))
59 oveq1 5550 . . . . 5 (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
6059eqeq1d 2090 . . . 4 (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
6160rspcev 2702 . . 3 ((𝑧 ∈ (2nd𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6258, 35, 61syl2anc 403 . 2 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6328, 62rexlimddv 2482 1 (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  {cab 2068  wrex 2350  cop 3409   class class class wbr 3793  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  Qcnq 6532   +Q cplq 6534   <Q cltq 6537  Pcnp 6543   +P cpp 6545  <P cltp 6547
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemexbt  6958
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