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Theorem ltrennb 7626
 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 7195 . . 3 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q ))
2 nnnq 7194 . . . . 5 (𝐽N → [⟨𝐽, 1o⟩] ~QQ)
32adantr 272 . . . 4 ((𝐽N𝐾N) → [⟨𝐽, 1o⟩] ~QQ)
4 nnnq 7194 . . . . 5 (𝐾N → [⟨𝐾, 1o⟩] ~QQ)
54adantl 273 . . . 4 ((𝐽N𝐾N) → [⟨𝐾, 1o⟩] ~QQ)
6 ltnqpr 7365 . . . 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 406 . . 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 7319 . . . . 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 7319 . . . . 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 7559 . . . 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 406 . . 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 213 . 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 7611 . 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 197 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 103   ↔ wb 104   ∈ wcel 1463  {cab 2101  ⟨cop 3498   class class class wbr 3897  (class class class)co 5740  1oc1o 6272  [cec 6393  Ncnpi 7044
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