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Theorem archrecpr 6819
Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
Assertion
Ref Expression
archrecpr (𝐴P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)
Distinct variable groups:   𝐴,𝑗   𝑗,𝑙,𝑢
Allowed substitution hints:   𝐴(𝑢,𝑙)

Proof of Theorem archrecpr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prop 6630 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 6632 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
31, 2syl 14 . . 3 (𝐴P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
4 archrecnq 6818 . . . . 5 (𝑥Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥)
54ad2antrl 467 . . . 4 ((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥)
61ad2antrr 465 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 simplrr 496 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → 𝑥 ∈ (1st𝐴))
8 prcdnql 6639 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴)))
96, 7, 8syl2anc 397 . . . . 5 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴)))
109reximdva 2438 . . . 4 ((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → (∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥 → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴)))
115, 10mpd 13 . . 3 ((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴))
123, 11rexlimddv 2454 . 2 (𝐴P → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴))
13 nnnq 6577 . . . . . 6 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
14 recclnq 6547 . . . . . 6 ([⟨𝑗, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
1513, 14syl 14 . . . . 5 (𝑗N → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
1615adantl 266 . . . 4 ((𝐴P𝑗N) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
17 simpl 106 . . . 4 ((𝐴P𝑗N) → 𝐴P)
18 nqprl 6706 . . . 4 (((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q𝐴P) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴) ↔ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
1916, 17, 18syl2anc 397 . . 3 ((𝐴P𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴) ↔ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
2019rexbidva 2340 . 2 (𝐴P → (∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴) ↔ ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
2112, 20mpbid 139 1 (𝐴P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  {cab 2042  wrex 2324  cop 3405   class class class wbr 3791  cfv 4929  1st c1st 5792  2nd c2nd 5793  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   ~Q ceq 6434  Qcnq 6435  *Qcrq 6439   <Q cltq 6440  Pcnp 6446  <P cltp 6450
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  caucvgprprlemlim  6866
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