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Theorem archpr 6930
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 6840. (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 6762 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6765 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
31, 2syl 14 . 2 (𝐴P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
4 archnqq 6704 . . . 4 (𝑧Q → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
54ad2antrl 474 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
6 simprl 498 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧Q)
76ad2antrr 472 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧Q)
8 simprr 499 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧 ∈ (2nd𝐴))
98ad2antrr 472 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ (2nd𝐴))
10 simpr 108 . . . . . . . 8 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
11 vex 2612 . . . . . . . . 9 𝑧 ∈ V
12 breq1 3808 . . . . . . . . 9 (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ))
13 ltnqex 6836 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q } ∈ V
14 gtnqex 6837 . . . . . . . . . 10 {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢} ∈ V
1513, 14op1st 5824 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) = {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }
1611, 12, 15elab2 2749 . . . . . . . 8 (𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
1710, 16sylibr 132 . . . . . . 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 2145 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (2nd𝐴) ↔ 𝑧 ∈ (2nd𝐴)))
19 eleq1 2145 . . . . . . . . 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 457 . . . . . . . 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 2709 . . . . . . 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 1168 . . . . . 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 500 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴P)
24 nnprlu 6840 . . . . . . . 8 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
2524ad2antlr 473 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6793 . . . . . . 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 403 . . . . . 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 165 . . . . 5 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
2928ex 113 . . . 4 (((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) → (𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))
3029reximdva 2468 . . 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 2486 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 102  wb 103  wcel 1434  {cab 2069  wrex 2354  cop 3419   class class class wbr 3805  cfv 4952  1st c1st 5816  2nd c2nd 5817  1𝑜c1o 6078  [cec 6191  Ncnpi 6559   ~Q ceq 6566  Qcnq 6567   <Q cltq 6572  Pcnp 6578  <P cltp 6582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2611  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6591  df-pli 6592  df-mi 6593  df-lti 6594  df-plpq 6631  df-mpq 6632  df-enq 6634  df-nqqs 6635  df-plqqs 6636  df-mqqs 6637  df-1nqqs 6638  df-rq 6639  df-ltnqqs 6640  df-inp 6753  df-iltp 6757
This theorem is referenced by:  archsr  7055
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