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Theorem archsr 6866
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 6812 . 2 R = ((P × P) / ~R )
2 breq1 3767 . . 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 2327 . 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 6652 . . . . . . 7 1PP
5 addclpr 6635 . . . . . . 7 ((𝑧P ∧ 1PP) → (𝑧 +P 1P) ∈ P)
64, 5mpan2 401 . . . . . 6 (𝑧P → (𝑧 +P 1P) ∈ P)
7 archpr 6741 . . . . . 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 261 . . . 4 ((𝑧P𝑤P) → ∃𝑥N (𝑧 +P 1P)<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
10 nnprlu 6651 . . . . . . . . . 10 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
1110adantl 262 . . . . . . . . 9 (((𝑧P𝑤P) ∧ 𝑥N) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
12 addclpr 6635 . . . . . . . . 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 392 . . . . . . . 8 (((𝑧P𝑤P) ∧ 𝑥N) → (⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
14 simplr 482 . . . . . . . 8 (((𝑧P𝑤P) ∧ 𝑥N) → 𝑤P)
15 ltaddpr 6695 . . . . . . . 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 391 . . . . . . 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 6676 . . . . . . . 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 391 . . . . . . 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 3790 . . . . . 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 6695 . . . . . . . 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 392 . . . . . . 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 6694 . . . . . . . . 9 <P Or P
23 ltrelpr 6603 . . . . . . . . 9 <P ⊆ (P × P)
2422, 23sotri 4720 . . . . . . . 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 (((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 4720 . . . . . . 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 (((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 2421 . . . 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 102 . . . . 5 (((𝑧P𝑤P) ∧ 𝑥N) → (𝑧P𝑤P))
334a1i 9 . . . . 5 (((𝑧P𝑤P) ∧ 𝑥N) → 1PP)
34 ltsrprg 6832 . . . . 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 1133 . . . 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 2323 . . 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 156 . 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 6193 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 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wrex 2307  cop 3378   class class class wbr 3764  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~Q ceq 6377   <Q cltq 6383  Pcnp 6389  1Pc1p 6390   +P cpp 6391  <P cltp 6393   ~R cer 6394  Rcnr 6395   <R cltr 6401
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815
This theorem is referenced by:  axarch  6965
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