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Theorem nqprl 6532
 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 ((A Q B P) → (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 6457 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
2 prnmaxl 6470 . . . . . 6 ((⟨(1stB), (2ndB)⟩ P A (1stB)) → x (1stB)A <Q x)
31, 2sylan 267 . . . . 5 ((B P A (1stB)) → x (1stB)A <Q x)
4 elprnql 6463 . . . . . . . . . 10 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → x Q)
51, 4sylan 267 . . . . . . . . 9 ((B P x (1stB)) → x Q)
65ad2ant2r 478 . . . . . . . 8 (((B P A (1stB)) (x (1stB) A <Q x)) → x Q)
7 vex 2554 . . . . . . . . . . . 12 x V
8 breq2 3759 . . . . . . . . . . . 12 (u = x → (A <Q uA <Q x))
97, 8elab 2681 . . . . . . . . . . 11 (x {uA <Q u} ↔ A <Q x)
109biimpri 124 . . . . . . . . . 10 (A <Q xx {uA <Q u})
11 ltnqex 6530 . . . . . . . . . . . 12 {𝑙𝑙 <Q A} V
12 gtnqex 6531 . . . . . . . . . . . 12 {uA <Q u} V
1311, 12op2nd 5716 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) = {uA <Q u}
1413eleq2i 2101 . . . . . . . . . 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 (((B P A (1stB)) (x (1stB) A <Q x)) → x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩))
17 simprl 483 . . . . . . . 8 (((B P A (1stB)) (x (1stB) A <Q x)) → x (1stB))
18 19.8a 1479 . . . . . . . 8 ((x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))) → x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
196, 16, 17, 18syl12anc 1132 . . . . . . 7 (((B P A (1stB)) (x (1stB) A <Q x)) → x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
20 df-rex 2306 . . . . . . 7 (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)) ↔ x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2119, 20sylibr 137 . . . . . 6 (((B P A (1stB)) (x (1stB) A <Q x)) → x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))
22 elprnql 6463 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P A (1stB)) → A Q)
231, 22sylan 267 . . . . . . . 8 ((B P A (1stB)) → A Q)
24 simpl 102 . . . . . . . 8 ((B P A (1stB)) → B P)
25 nqprlu 6529 . . . . . . . . 9 (A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
26 ltdfpr 6488 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2725, 26sylan 267 . . . . . . . 8 ((A Q B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2823, 24, 27syl2anc 391 . . . . . . 7 ((B P A (1stB)) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2928adantr 261 . . . . . 6 (((B P A (1stB)) (x (1stB) A <Q x)) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
3021, 29mpbird 156 . . . . 5 (((B P A (1stB)) (x (1stB) A <Q x)) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B)
313, 30rexlimddv 2431 . . . 4 ((B P A (1stB)) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B)
3231ex 108 . . 3 (B P → (A (1stB) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))
3332adantl 262 . 2 ((A Q B P) → (A (1stB) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))
3427biimpa 280 . . . 4 (((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) → x Q (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 ((x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))) → A <Q x)
3837adantl 262 . . . . 5 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → A <Q x)
39 simpllr 486 . . . . . 6 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → B P)
40 simprrr 492 . . . . . 6 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → x (1stB))
41 prcdnql 6466 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → (A <Q xA (1stB)))
421, 41sylan 267 . . . . . 6 ((B P x (1stB)) → (A <Q xA (1stB)))
4339, 40, 42syl2anc 391 . . . . 5 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → (A <Q xA (1stB)))
4438, 43mpd 13 . . . 4 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → A (1stB))
4534, 44rexlimddv 2431 . . 3 (((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) → A (1stB))
4645ex 108 . 2 ((A Q B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P BA (1stB)))
4733, 46impbid 120 1 ((A Q B P) → (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 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264
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