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Theorem archsr 6668
Description: For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R is the embedding of the positive integer x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
Assertion
Ref Expression
archsr (A Rx N A <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R )
Distinct variable group:   A,𝑙,u,x

Proof of Theorem archsr
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 6615 . 2 R = ((P × P) / ~R )
2 breq1 3758 . . 3 ([⟨z, w⟩] ~R = A → ([⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~RA <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ))
32rexbidv 2321 . 2 ([⟨z, w⟩] ~R = A → (x N [⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~Rx N A <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ))
4 1pr 6534 . . . . . . 7 1P P
5 addclpr 6520 . . . . . . 7 ((z P 1P P) → (z +P 1P) P)
64, 5mpan2 401 . . . . . 6 (z P → (z +P 1P) P)
7 archpr 6613 . . . . . 6 ((z +P 1P) Px N (z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
86, 7syl 14 . . . . 5 (z Px N (z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
98adantr 261 . . . 4 ((z P w P) → x N (z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
10 nnprlu 6533 . . . . . . . . . 10 (x N → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P)
1110adantl 262 . . . . . . . . 9 (((z P w P) x N) → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P)
12 addclpr 6520 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P 1P P) → (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P)
1311, 4, 12sylancl 392 . . . . . . . 8 (((z P w P) x N) → (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P)
14 simplr 482 . . . . . . . 8 (((z P w P) x N) → w P)
15 ltaddpr 6569 . . . . . . . 8 (((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P w P) → (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)<P ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) +P w))
1613, 14, 15syl2anc 391 . . . . . . 7 (((z P w P) x N) → (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)<P ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) +P w))
17 addcomprg 6552 . . . . . . . 8 ((w P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P) → (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)) = ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) +P w))
1814, 13, 17syl2anc 391 . . . . . . 7 (((z P w P) x N) → (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)) = ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) +P w))
1916, 18breqtrrd 3781 . . . . . 6 (((z P w P) x N) → (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)))
20 ltaddpr 6569 . . . . . . . 8 ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P 1P P) → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))
2111, 4, 20sylancl 392 . . . . . . 7 (((z P w P) x N) → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))
22 ltsopr 6568 . . . . . . . . 9 <P Or P
23 ltrelpr 6487 . . . . . . . . 9 <P ⊆ (P × P)
2422, 23sotri 4663 . . . . . . . 8 (((z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)) → (z +P 1P)<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))
2524expcom 109 . . . . . . 7 (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) → ((z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ → (z +P 1P)<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)))
2621, 25syl 14 . . . . . 6 (((z P w P) x N) → ((z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ → (z +P 1P)<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)))
2722, 23sotri 4663 . . . . . . 7 (((z +P 1P)<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))) → (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)))
2827expcom 109 . . . . . 6 ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)) → ((z +P 1P)<P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) → (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
2919, 26, 28sylsyld 52 . . . . 5 (((z P w P) x N) → ((z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ → (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
3029reximdva 2415 . . . 4 ((z P w P) → (x N (z +P 1P)<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ → x N (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
319, 30mpd 13 . . 3 ((z P w P) → x N (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P)))
32 simpl 102 . . . . 5 (((z P w P) x N) → (z P w P))
334a1i 9 . . . . 5 (((z P w P) x N) → 1P P)
34 ltsrprg 6635 . . . . 5 (((z P w P) ((⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P 1P P)) → ([⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ↔ (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
3532, 13, 33, 34syl12anc 1132 . . . 4 (((z P w P) x N) → ([⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ↔ (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
3635rexbidva 2317 . . 3 ((z P w P) → (x N [⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~Rx N (z +P 1P)<P (w +P (⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P))))
3731, 36mpbird 156 . 2 ((z P w P) → x N [⟨z, w⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R )
381, 3, 37ecoptocl 6129 1 (A Rx N A <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R )
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  cop 3370   class class class wbr 3755  (class class class)co 5455  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   ~Q ceq 6263   <Q cltq 6269  Pcnp 6275  1Pc1p 6276   +P cpp 6277  <P cltp 6279   ~R cer 6280  Rcnr 6281   <R cltr 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6614  df-nr 6615  df-ltr 6618
This theorem is referenced by:  axarch  6733
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