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Theorem archrecpr 7144
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 6955 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 6957 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
31, 2syl 14 . . 3 (𝐴P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
4 archrecnq 7143 . . . . 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 6964 . . . . . 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 2471 . . . 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 2489 . 2 (𝐴P → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) ∈ (1st𝐴))
13 nnnq 6902 . . . . . 6 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
14 recclnq 6872 . . . . . 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 7031 . . . 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 2373 . 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 1436  {cab 2071  wrex 2356  cop 3428   class class class wbr 3814  cfv 4972  1st c1st 5847  2nd c2nd 5848  1𝑜c1o 6109  [cec 6223  Ncnpi 6752   ~Q ceq 6759  Qcnq 6760  *Qcrq 6764   <Q cltq 6765  Pcnp 6771  <P cltp 6775
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-inp 6946  df-iltp 6950
This theorem is referenced by:  caucvgprprlemlim  7191
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