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Theorem archpr 7465
Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 7375. (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
archpr (𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
Distinct variable group:   𝐴,𝑙,𝑢,𝑥

Proof of Theorem archpr
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7297 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7300 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
31, 2syl 14 . 2 (𝐴P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
4 archnqq 7239 . . . 4 (𝑧Q → ∃𝑥N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
54ad2antrl 481 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
6 simprl 520 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧Q)
76ad2antrr 479 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧Q)
8 simprr 521 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧 ∈ (2nd𝐴))
98ad2antrr 479 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ (2nd𝐴))
10 simpr 109 . . . . . . . 8 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
11 vex 2689 . . . . . . . . 9 𝑧 ∈ V
12 breq1 3932 . . . . . . . . 9 (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13 ltnqex 7371 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14 gtnqex 7372 . . . . . . . . . 10 {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
1513, 14op1st 6044 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) = {𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }
1611, 12, 15elab2 2832 . . . . . . . 8 (𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
1710, 16sylibr 133 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
18 eleq1 2202 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (2nd𝐴) ↔ 𝑧 ∈ (2nd𝐴)))
19 eleq1 2202 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
2018, 19anbi12d 464 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)) ↔ (𝑧 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
2120rspcev 2789 . . . . . . 7 ((𝑧Q ∧ (𝑧 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))) → ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
227, 9, 17, 21syl12anc 1214 . . . . . 6 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
23 simplll 522 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴P)
24 nnprlu 7375 . . . . . . . 8 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
2524ad2antlr 480 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
26 ltdfpr 7328 . . . . . . 7 ((𝐴P ∧ ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
2723, 25, 26syl2anc 408 . . . . . 6 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → (𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
2822, 27mpbird 166 . . . . 5 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
2928ex 114 . . . 4 (((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) → (𝑧 <Q [⟨𝑥, 1o⟩] ~Q𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
3029reximdva 2534 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → (∃𝑥N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
315, 30mpd 13 . 2 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
323, 31rexlimddv 2554 1 (𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  {cab 2125  wrex 2417  cop 3530   class class class wbr 3929  cfv 5123  1st c1st 6036  2nd c2nd 6037  1oc1o 6306  [cec 6427  Ncnpi 7094   ~Q ceq 7101  Qcnq 7102   <Q cltq 7107  Pcnp 7113  <P cltp 7117
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-inp 7288  df-iltp 7292
This theorem is referenced by:  archsr  7604
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