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Theorem archsr 7744
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 (𝐴R → ∃𝑥N 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 7689 . 2 R = ((P × P) / ~R )
2 breq1 3992 . . 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 2471 . 2 ([⟨𝑧, 𝑤⟩] ~R = 𝐴 → (∃𝑥N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ ∃𝑥N 𝐴 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
4 1pr 7516 . . . . . . 7 1PP
5 addclpr 7499 . . . . . . 7 ((𝑧P ∧ 1PP) → (𝑧 +P 1P) ∈ P)
64, 5mpan2 423 . . . . . 6 (𝑧P → (𝑧 +P 1P) ∈ P)
7 archpr 7605 . . . . . 6 ((𝑧 +P 1P) ∈ P → ∃𝑥N (𝑧 +P 1P)<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
86, 7syl 14 . . . . 5 (𝑧P → ∃𝑥N (𝑧 +P 1P)<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
98adantr 274 . . . 4 ((𝑧P𝑤P) → ∃𝑥N (𝑧 +P 1P)<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
10 nnprlu 7515 . . . . . . . . . 10 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
1110adantl 275 . . . . . . . . 9 (((𝑧P𝑤P) ∧ 𝑥N) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
12 addclpr 7499 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1PP) → (⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
1311, 4, 12sylancl 411 . . . . . . . 8 (((𝑧P𝑤P) ∧ 𝑥N) → (⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
14 simplr 525 . . . . . . . 8 (((𝑧P𝑤P) ∧ 𝑥N) → 𝑤P)
15 ltaddpr 7559 . . . . . . . 8 (((⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P𝑤P) → (⟨{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 409 . . . . . . 7 (((𝑧P𝑤P) ∧ 𝑥N) → (⟨{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 7540 . . . . . . . 8 ((𝑤P ∧ (⟨{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 409 . . . . . . 7 (((𝑧P𝑤P) ∧ 𝑥N) → (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 4017 . . . . . 6 (((𝑧P𝑤P) ∧ 𝑥N) → (⟨{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 7559 . . . . . . . 8 ((⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1PP) → ⟨{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 411 . . . . . . 7 (((𝑧P𝑤P) ∧ 𝑥N) → ⟨{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 7558 . . . . . . . . 9 <P Or P
23 ltrelpr 7467 . . . . . . . . 9 <P ⊆ (P × P)
2422, 23sotri 5006 . . . . . . . 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 115 . . . . . . 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 (((𝑧P𝑤P) ∧ 𝑥N) → ((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 5006 . . . . . . 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 115 . . . . . 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 58 . . . . 5 (((𝑧P𝑤P) ∧ 𝑥N) → ((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 2572 . . . 4 ((𝑧P𝑤P) → (∃𝑥N (𝑧 +P 1P)<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ → ∃𝑥N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
319, 30mpd 13 . . 3 ((𝑧P𝑤P) → ∃𝑥N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
32 simpl 108 . . . . 5 (((𝑧P𝑤P) ∧ 𝑥N) → (𝑧P𝑤P))
334a1i 9 . . . . 5 (((𝑧P𝑤P) ∧ 𝑥N) → 1PP)
34 ltsrprg 7709 . . . . 5 (((𝑧P𝑤P) ∧ ((⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP)) → ([⟨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 1231 . . . 4 (((𝑧P𝑤P) ∧ 𝑥N) → ([⟨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 2467 . . 3 ((𝑧P𝑤P) → (∃𝑥N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ ∃𝑥N (𝑧 +P 1P)<P (𝑤 +P (⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
3731, 36mpbird 166 . 2 ((𝑧P𝑤P) → ∃𝑥N [⟨𝑧, 𝑤⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
381, 3, 37ecoptocl 6600 1 (𝐴R → ∃𝑥N 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 103  wb 104   = wceq 1348  wcel 2141  {cab 2156  wrex 2449  cop 3586   class class class wbr 3989  (class class class)co 5853  1oc1o 6388  [cec 6511  Ncnpi 7234   ~Q ceq 7241   <Q cltq 7247  Pcnp 7253  1Pc1p 7254   +P cpp 7255  <P cltp 7257   ~R cer 7258  Rcnr 7259   <R cltr 7265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692
This theorem is referenced by:  axarch  7853
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