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Theorem prplnqu 6718
 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 6645 . . . . . . . 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 6695 . . . . . . 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 391 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋))
7 addcomprg 6676 . . . . . . 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 391 . . . . . 6 (𝜑 → (⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
96, 8breqtrd 3788 . . . . 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 6635 . . . . . . . . 9 ((𝑋P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
124, 3, 11syl2anc 391 . . . . . . . 8 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P)
13 prop 6573 . . . . . . . . 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 6580 . . . . . . . . 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 267 . . . . . . . 8 (((𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ∈ P𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))) → 𝐴Q)
1612, 10, 15syl2anc 391 . . . . . . 7 (𝜑𝐴Q)
17 nqpru 6650 . . . . . . 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 391 . . . . . 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 135 . . . . 5 (𝜑 → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
20 ltsopr 6694 . . . . . 6 <P Or P
21 ltrelpr 6603 . . . . . 6 <P ⊆ (P × P)
2220, 21sotri 4720 . . . . 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 391 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
24 ltnqpr 6691 . . . . 5 ((𝑄Q𝐴Q) → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
251, 16, 24syl2anc 391 . . . 4 (𝜑 → (𝑄 <Q 𝐴 ↔ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
2623, 25mpbird 156 . . 3 (𝜑𝑄 <Q 𝐴)
27 ltexnqi 6507 . . 3 (𝑄 <Q 𝐴 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2826, 27syl 14 . 2 (𝜑 → ∃𝑧Q (𝑄 +Q 𝑧) = 𝐴)
2919adantr 261 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
301adantr 261 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄Q)
31 simprl 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧Q)
32 addcomnqg 6479 . . . . . . . . . 10 ((𝑄Q𝑧Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
3330, 31, 32syl2anc 391 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34 simprr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
3533, 34eqtr3d 2074 . . . . . . . 8 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36 breq2 3768 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
3736abbidv 2155 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑙𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙𝑙 <Q 𝐴})
38 breq1 3767 . . . . . . . . . 10 ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢𝐴 <Q 𝑢))
3938abbidv 2155 . . . . . . . . 9 ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢𝐴 <Q 𝑢})
4037, 39opeq12d 3557 . . . . . . . 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 6659 . . . . . . . 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 391 . . . . . . 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 2074 . . . . . 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 3788 . . . . 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 6717 . . . . . . 7 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
4746adantl 262 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
484adantr 261 . . . . . 6 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋P)
49 nqprlu 6645 . . . . . . 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 6676 . . . . . . 7 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5352adantl 262 . . . . . 6 (((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5447, 48, 50, 51, 53caovord2d 5670 . . . . 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 156 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩)
56 nqpru 6650 . . . . 5 ((𝑧Q𝑋P) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5731, 48, 56syl2anc 391 . . . 4 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 ∈ (2nd𝑋) ↔ 𝑋<P ⟨{𝑙𝑙 <Q 𝑧}, {𝑢𝑧 <Q 𝑢}⟩))
5855, 57mpbird 156 . . 3 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd𝑋))
59 oveq1 5519 . . . . 5 (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
6059eqeq1d 2048 . . . 4 (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
6160rspcev 2656 . . 3 ((𝑧 ∈ (2nd𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6258, 35, 61syl2anc 391 . 2 ((𝜑 ∧ (𝑧Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
6328, 62rexlimddv 2437 1 (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6378   +Q cplq 6380
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