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Theorem prplnqu 7619
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 7546 . . . . . . . 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 7596 . . . . . . 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 411 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋))
7 addcomprg 7577 . . . . . . 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 411 . . . . . 6 (𝜑 → (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
96, 8breqtrd 4030 . . . . 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 7536 . . . . . . . . 9 ((𝑋P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
124, 3, 11syl2anc 411 . . . . . . . 8 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
13 prop 7474 . . . . . . . . 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 7481 . . . . . . . . 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 283 . . . . . . . 8 (((𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))) → 𝐴Q)
1612, 10, 15syl2anc 411 . . . . . . 7 (𝜑𝐴Q)
17 nqpru 7551 . . . . . . 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 411 . . . . . 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 147 . . . . 5 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
20 ltsopr 7595 . . . . . 6 <P Or P
21 ltrelpr 7504 . . . . . 6 <P ⊆ (P × P)
2220, 21sotri 5025 . . . . 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 411 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
24 ltnqpr 7592 . . . . 5 ((𝑄Q𝐴Q) → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
251, 16, 24syl2anc 411 . . . 4 (𝜑 → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
2623, 25mpbird 167 . . 3 (𝜑𝑄 <Q 𝐴)
27 ltexnqi 7408 . . 3 (𝑄 <Q 𝐴 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2826, 27syl 14 . 2 (𝜑 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2919adantr 276 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
301adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄Q)
31 simprl 529 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧Q)
32 addcomnqg 7380 . . . . . . . . . 10 ((𝑄Q𝑧Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
3330, 31, 32syl2anc 411 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34 simprr 531 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
3533, 34eqtr3d 2212 . . . . . . . 8 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36 breq2 4008 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
3736abbidv 2295 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑙𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙𝑙 <Q 𝐴})
38 breq1 4007 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢𝐴 <Q 𝑢))
3938abbidv 2295 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢𝐴 <Q 𝑢})
4037, 39opeq12d 3787 . . . . . . . 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 7560 . . . . . . . 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 411 . . . . . . 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 2212 . . . . . 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 4030 . . . . 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 7618 . . . . . . 7 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
4746adantl 277 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
484adantr 276 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋P)
49 nqprlu 7546 . . . . . . 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 7577 . . . . . . 7 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5352adantl 277 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5447, 48, 50, 51, 53caovord2d 6044 . . . . 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 167 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩)
56 nqpru 7551 . . . . 5 ((𝑧Q𝑋P) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5731, 48, 56syl2anc 411 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5855, 57mpbird 167 . . 3 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd𝑋))
59 oveq1 5882 . . . . 5 (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
6059eqeq1d 2186 . . . 4 (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
6160rspcev 2842 . . 3 ((𝑧 ∈ (2nd𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6258, 35, 61syl2anc 411 . 2 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6328, 62rexlimddv 2599 1 (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  {cab 2163  wrex 2456  cop 3596   class class class wbr 4004  cfv 5217  (class class class)co 5875  1st c1st 6139  2nd c2nd 6140  Qcnq 7279   +Q cplq 7281   <Q cltq 7284  Pcnp 7290   +P cpp 7292  <P cltp 7294
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469
This theorem is referenced by:  caucvgprprlemexbt  7705
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