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Theorem nqprl 6649
 Description: Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by
Assertion
Ref Expression
nqprl ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
Distinct variable group:   𝐴,𝑙,𝑢
Allowed substitution hints:   𝐵(𝑢,𝑙)

Proof of Theorem nqprl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaxl 6586 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
31, 2sylan 267 . . . . 5 ((𝐵P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
4 elprnql 6579 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → 𝑥Q)
51, 4sylan 267 . . . . . . . . 9 ((𝐵P𝑥 ∈ (1st𝐵)) → 𝑥Q)
65ad2ant2r 478 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥Q)
7 vex 2560 . . . . . . . . . . . 12 𝑥 ∈ V
8 breq2 3768 . . . . . . . . . . . 12 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
97, 8elab 2687 . . . . . . . . . . 11 (𝑥 ∈ {𝑢𝐴 <Q 𝑢} ↔ 𝐴 <Q 𝑥)
109biimpri 124 . . . . . . . . . 10 (𝐴 <Q 𝑥𝑥 ∈ {𝑢𝐴 <Q 𝑢})
11 ltnqex 6647 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
12 gtnqex 6648 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1311, 12op2nd 5774 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1413eleq2i 2104 . . . . . . . . . 10 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑢𝐴 <Q 𝑢})
1510, 14sylibr 137 . . . . . . . . 9 (𝐴 <Q 𝑥𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1615ad2antll 460 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
17 simprl 483 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (1st𝐵))
18 19.8a 1482 . . . . . . . 8 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
196, 16, 17, 18syl12anc 1133 . . . . . . 7 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
20 df-rex 2312 . . . . . . 7 (∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)) ↔ ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2119, 20sylibr 137 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
22 elprnql 6579 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → 𝐴Q)
231, 22sylan 267 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐴Q)
24 simpl 102 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐵P)
25 nqprlu 6645 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6604 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2725, 26sylan 267 . . . . . . . 8 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2823, 24, 27syl2anc 391 . . . . . . 7 ((𝐵P𝐴 ∈ (1st𝐵)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2928adantr 261 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
3021, 29mpbird 156 . . . . 5 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
313, 30rexlimddv 2437 . . . 4 ((𝐵P𝐴 ∈ (1st𝐵)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
3231ex 108 . . 3 (𝐵P → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3332adantl 262 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3427biimpa 280 . . . 4 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
3514, 9bitri 173 . . . . . . . 8 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
3635biimpi 113 . . . . . . 7 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑥)
3736ad2antrl 459 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → 𝐴 <Q 𝑥)
3837adantl 262 . . . . 5 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐴 <Q 𝑥)
39 simpllr 486 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐵P)
40 simprrr 492 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (1st𝐵))
41 prcdnql 6582 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
421, 41sylan 267 . . . . . 6 ((𝐵P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4339, 40, 42syl2anc 391 . . . . 5 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4438, 43mpd 13 . . . 4 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐴 ∈ (1st𝐵))
4534, 44rexlimddv 2437 . . 3 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → 𝐴 ∈ (1st𝐵))
4645ex 108 . 2 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵𝐴 ∈ (1st𝐵)))
4733, 46impbid 120 1 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6378
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