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Theorem ltrennb 6988
 Description: Ordering of natural numbers with
Assertion
Ref Expression
ltrennb ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
Distinct variable groups:   𝐽,𝑙   𝑢,𝐽   𝐾,𝑙   𝑢,𝐾

Proof of Theorem ltrennb
StepHypRef Expression
1 ltnnnq 6579 . . 3 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q ))
2 nnnq 6578 . . . . 5 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
32adantr 265 . . . 4 ((𝐽N𝐾N) → [⟨𝐽, 1𝑜⟩] ~QQ)
4 nnnq 6578 . . . . 5 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
54adantl 266 . . . 4 ((𝐽N𝐾N) → [⟨𝐾, 1𝑜⟩] ~QQ)
6 ltnqpr 6749 . . . 4 (([⟨𝐽, 1𝑜⟩] ~QQ ∧ [⟨𝐾, 1𝑜⟩] ~QQ) → ([⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q ↔ ⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩))
73, 5, 6syl2anc 397 . . 3 ((𝐽N𝐾N) → ([⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q ↔ ⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩))
8 nqprlu 6703 . . . . 5 ([⟨𝐽, 1𝑜⟩] ~QQ → ⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
93, 8syl 14 . . . 4 ((𝐽N𝐾N) → ⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
10 nqprlu 6703 . . . . 5 ([⟨𝐾, 1𝑜⟩] ~QQ → ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
115, 10syl 14 . . . 4 ((𝐽N𝐾N) → ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
12 prsrlt 6929 . . . 4 ((⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P) → (⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ↔ [⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
139, 11, 12syl2anc 397 . . 3 ((𝐽N𝐾N) → (⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ↔ [⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
141, 7, 133bitrd 207 . 2 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ [⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
15 ltresr 6973 . 2 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ [⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1614, 15syl6bbr 191 1 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∈ wcel 1409  {cab 2042  ⟨cop 3406   class class class wbr 3792  (class class class)co 5540  1𝑜c1o 6025  [cec 6135  Ncnpi 6428
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