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Theorem ltrennb 7871
Description: Ordering of natural numbers with <N or <. (Contributed by Jim Kingdon, 13-Jul-2021.)
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 7440 . . 3 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q ))
2 nnnq 7439 . . . . 5 (𝐽N → [⟨𝐽, 1o⟩] ~QQ)
32adantr 276 . . . 4 ((𝐽N𝐾N) → [⟨𝐽, 1o⟩] ~QQ)
4 nnnq 7439 . . . . 5 (𝐾N → [⟨𝐾, 1o⟩] ~QQ)
54adantl 277 . . . 4 ((𝐽N𝐾N) → [⟨𝐾, 1o⟩] ~QQ)
6 ltnqpr 7610 . . . 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 411 . . 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 7564 . . . . 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 7564 . . . . 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 7804 . . . 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 411 . . 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 214 . 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 7856 . 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, 15bitr4di 198 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 104  wb 105  wcel 2160  {cab 2175  cop 3610   class class class wbr 4018  (class class class)co 5891  1oc1o 6428  [cec 6551  Ncnpi 7289   <N clti 7292   ~Q ceq 7296  Qcnq 7297   <Q cltq 7302  Pcnp 7308  1Pc1p 7309   +P cpp 7310  <P cltp 7312   ~R cer 7313  0Rc0r 7315   <R cltr 7320   < cltrr 7833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-i1p 7484  df-iplp 7485  df-iltp 7487  df-enr 7743  df-nr 7744  df-ltr 7747  df-0r 7748  df-r 7839  df-lt 7842
This theorem is referenced by:  ltrenn  7872  axcaucvglemres  7916
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