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Theorem archpr 6769
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 6679. (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 6601 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6604 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
31, 2syl 14 . 2 (𝐴P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
4 archnqq 6543 . . . 4 (𝑧Q → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
54ad2antrl 467 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
6 simprl 491 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧Q)
76ad2antrr 465 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧Q)
8 simprr 492 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧 ∈ (2nd𝐴))
98ad2antrr 465 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ (2nd𝐴))
10 simpr 107 . . . . . . . 8 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
11 vex 2575 . . . . . . . . 9 𝑧 ∈ V
12 breq1 3792 . . . . . . . . 9 (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ))
13 ltnqex 6675 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q } ∈ V
14 gtnqex 6676 . . . . . . . . . 10 {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢} ∈ V
1513, 14op1st 5798 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) = {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }
1611, 12, 15elab2 2710 . . . . . . . 8 (𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
1710, 16sylibr 141 . . . . . . 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 2114 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (2nd𝐴) ↔ 𝑧 ∈ (2nd𝐴)))
19 eleq1 2114 . . . . . . . . 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 450 . . . . . . . 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 2671 . . . . . . 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 1142 . . . . . 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 493 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴P)
24 nnprlu 6679 . . . . . . . 8 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
2524ad2antlr 466 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6632 . . . . . . 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 397 . . . . . 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 160 . . . . 5 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
2928ex 112 . . . 4 (((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) → (𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))
3029reximdva 2436 . . 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 2452 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 101  wb 102  wcel 1407  {cab 2040  wrex 2322  cop 3403   class class class wbr 3789  cfv 4927  1st c1st 5790  2nd c2nd 5791  1𝑜c1o 6022  [cec 6132  Ncnpi 6398   ~Q ceq 6405  Qcnq 6406   <Q cltq 6411  Pcnp 6417  <P cltp 6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-inp 6592  df-iltp 6596
This theorem is referenced by:  archsr  6894
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