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Theorem ltnqpri 6898
 Description: We can order fractions via
Assertion
Ref Expression
ltnqpri (𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)
Distinct variable groups:   𝐴,𝑙   𝑢,𝐴   𝐵,𝑙   𝑢,𝐵

Proof of Theorem ltnqpri
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6669 . . . . . . . 8 <Q ⊆ (Q × Q)
21brel 4438 . . . . . . 7 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
32simpld 110 . . . . . 6 (𝐴 <Q 𝐵𝐴Q)
4 nqprlu 6851 . . . . . 6 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
62simprd 112 . . . . . 6 (𝐴 <Q 𝐵𝐵Q)
7 nqprlu 6851 . . . . . 6 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . 5 (𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
9 ltdfpr 6810 . . . . 5 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
105, 8, 9syl2anc 403 . . . 4 (𝐴 <Q 𝐵 → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
11 vex 2613 . . . . . . 7 𝑥 ∈ V
12 breq2 3809 . . . . . . 7 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
13 ltnqex 6853 . . . . . . . 8 {𝑙𝑙 <Q 𝐴} ∈ V
14 gtnqex 6854 . . . . . . . 8 {𝑢𝐴 <Q 𝑢} ∈ V
1513, 14op2nd 5825 . . . . . . 7 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1611, 12, 15elab2 2749 . . . . . 6 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
17 breq1 3808 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q 𝐵𝑥 <Q 𝐵))
18 ltnqex 6853 . . . . . . . 8 {𝑙𝑙 <Q 𝐵} ∈ V
19 gtnqex 6854 . . . . . . . 8 {𝑢𝐵 <Q 𝑢} ∈ V
2018, 19op1st 5824 . . . . . . 7 (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐵}
2111, 17, 20elab2 2749 . . . . . 6 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐵)
2216, 21anbi12i 448 . . . . 5 ((𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ (𝐴 <Q 𝑥𝑥 <Q 𝐵))
2322rexbii 2378 . . . 4 (∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑥Q (𝐴 <Q 𝑥𝑥 <Q 𝐵))
2410, 23syl6bb 194 . . 3 (𝐴 <Q 𝐵 → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝐴 <Q 𝑥𝑥 <Q 𝐵)))
25 ltbtwnnqq 6719 . . 3 (𝐴 <Q 𝐵 ↔ ∃𝑥Q (𝐴 <Q 𝑥𝑥 <Q 𝐵))
2624, 25syl6bbr 196 . 2 (𝐴 <Q 𝐵 → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ↔ 𝐴 <Q 𝐵))
2726ibir 175 1 (𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∈ wcel 1434  {cab 2069  ∃wrex 2354  ⟨cop 3419   class class class wbr 3805  ‘cfv 4952  1st c1st 5816  2nd c2nd 5817  Qcnq 6584
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