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Theorem archpr 7642
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 7552. (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 7474 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7477 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
31, 2syl 14 . 2 (𝐴P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
4 archnqq 7416 . . . 4 (𝑧Q → ∃𝑥N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
54ad2antrl 490 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
6 simprl 529 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧Q)
76ad2antrr 488 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧Q)
8 simprr 531 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧 ∈ (2nd𝐴))
98ad2antrr 488 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ (2nd𝐴))
10 simpr 110 . . . . . . . 8 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
11 vex 2741 . . . . . . . . 9 𝑧 ∈ V
12 breq1 4007 . . . . . . . . 9 (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13 ltnqex 7548 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14 gtnqex 7549 . . . . . . . . . 10 {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
1513, 14op1st 6147 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) = {𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }
1611, 12, 15elab2 2886 . . . . . . . 8 (𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
1710, 16sylibr 134 . . . . . . 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 2240 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (2nd𝐴) ↔ 𝑧 ∈ (2nd𝐴)))
19 eleq1 2240 . . . . . . . . 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 473 . . . . . . . 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 2842 . . . . . . 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 1236 . . . . . 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 533 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴P)
24 nnprlu 7552 . . . . . . . 8 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
2524ad2antlr 489 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
26 ltdfpr 7505 . . . . . . 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 411 . . . . . 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 167 . . . . 5 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
2928ex 115 . . . 4 (((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) → (𝑧 <Q [⟨𝑥, 1o⟩] ~Q𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
3029reximdva 2579 . . 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 2599 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 104  wb 105  wcel 2148  {cab 2163  wrex 2456  cop 3596   class class class wbr 4004  cfv 5217  1st c1st 6139  2nd c2nd 6140  1oc1o 6410  [cec 6533  Ncnpi 7271   ~Q ceq 7278  Qcnq 7279   <Q cltq 7284  Pcnp 7290  <P cltp 7294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465  df-iltp 7469
This theorem is referenced by:  archsr  7781
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