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Theorem prplnqu 7435
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 7362 . . . . . . . 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 7412 . . . . . . 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 408 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋))
7 addcomprg 7393 . . . . . . 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 408 . . . . . 6 (𝜑 → (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
96, 8breqtrd 3954 . . . . 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 7352 . . . . . . . . 9 ((𝑋P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
124, 3, 11syl2anc 408 . . . . . . . 8 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
13 prop 7290 . . . . . . . . 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 7297 . . . . . . . . 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 281 . . . . . . . 8 (((𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))) → 𝐴Q)
1612, 10, 15syl2anc 408 . . . . . . 7 (𝜑𝐴Q)
17 nqpru 7367 . . . . . . 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 408 . . . . . 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 146 . . . . 5 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
20 ltsopr 7411 . . . . . 6 <P Or P
21 ltrelpr 7320 . . . . . 6 <P ⊆ (P × P)
2220, 21sotri 4934 . . . . 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 408 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
24 ltnqpr 7408 . . . . 5 ((𝑄Q𝐴Q) → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
251, 16, 24syl2anc 408 . . . 4 (𝜑 → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
2623, 25mpbird 166 . . 3 (𝜑𝑄 <Q 𝐴)
27 ltexnqi 7224 . . 3 (𝑄 <Q 𝐴 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2826, 27syl 14 . 2 (𝜑 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2919adantr 274 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
301adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄Q)
31 simprl 520 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧Q)
32 addcomnqg 7196 . . . . . . . . . 10 ((𝑄Q𝑧Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
3330, 31, 32syl2anc 408 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34 simprr 521 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
3533, 34eqtr3d 2174 . . . . . . . 8 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36 breq2 3933 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
3736abbidv 2257 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑙𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙𝑙 <Q 𝐴})
38 breq1 3932 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢𝐴 <Q 𝑢))
3938abbidv 2257 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢𝐴 <Q 𝑢})
4037, 39opeq12d 3713 . . . . . . . 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 7376 . . . . . . . 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 408 . . . . . . 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 2174 . . . . . 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 3954 . . . . 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 7434 . . . . . . 7 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
4746adantl 275 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
484adantr 274 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋P)
49 nqprlu 7362 . . . . . . 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 7393 . . . . . . 7 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5352adantl 275 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5447, 48, 50, 51, 53caovord2d 5940 . . . . 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 166 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩)
56 nqpru 7367 . . . . 5 ((𝑧Q𝑋P) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5731, 48, 56syl2anc 408 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5855, 57mpbird 166 . . 3 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd𝑋))
59 oveq1 5781 . . . . 5 (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
6059eqeq1d 2148 . . . 4 (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
6160rspcev 2789 . . 3 ((𝑧 ∈ (2nd𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6258, 35, 61syl2anc 408 . 2 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6328, 62rexlimddv 2554 1 (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  {cab 2125  wrex 2417  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097   <Q cltq 7100  Pcnp 7106   +P cpp 7108  <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  caucvgprprlemexbt  7521
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