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Theorem archrecpr 7202
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 7013 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 7015 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
31, 2syl 14 . . 3 (𝐴P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
4 archrecnq 7201 . . . . 5 (𝑥Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥)
54ad2antrl 474 . . . 4 ((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥)
61ad2antrr 472 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 simplrr 503 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → 𝑥 ∈ (1st𝐴))
8 prcdnql 7022 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴)))
96, 7, 8syl2anc 403 . . . . 5 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴)))
109reximdva 2475 . . . 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 2493 . 2 (𝐴P → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴))
13 nnnq 6960 . . . . . 6 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
14 recclnq 6930 . . . . . 6 ([⟨𝑗, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
1513, 14syl 14 . . . . 5 (𝑗N → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
1615adantl 271 . . . 4 ((𝐴P𝑗N) → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ Q)
17 simpl 107 . . . 4 ((𝐴P𝑗N) → 𝐴P)
18 nqprl 7089 . . . 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 403 . . 3 ((𝐴P𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴) ↔ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
2019rexbidva 2377 . 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 145 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 102  wb 103  wcel 1438  {cab 2074  wrex 2360  cop 3444   class class class wbr 3837  cfv 5002  1st c1st 5891  2nd c2nd 5892  1𝑜c1o 6156  [cec 6270  Ncnpi 6810   ~Q ceq 6817  Qcnq 6818  *Qcrq 6822   <Q cltq 6823  Pcnp 6829  <P cltp 6833
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004  df-iltp 7008
This theorem is referenced by:  caucvgprprlemlim  7249
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