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Theorem archrecpr 7676
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 7487 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 7489 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
31, 2syl 14 . . 3 (𝐴P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
4 archrecnq 7675 . . . . 5 (𝑥Q → ∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
54ad2antrl 490 . . . 4 ((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
61ad2antrr 488 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 simplrr 536 . . . . . 6 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → 𝑥 ∈ (1st𝐴))
8 prcdnql 7496 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st𝐴)))
96, 7, 8syl2anc 411 . . . . 5 (((𝐴P ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑗N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st𝐴)))
109reximdva 2589 . . . 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 2609 . 2 (𝐴P → ∃𝑗N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st𝐴))
13 nnnq 7434 . . . . . 6 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
14 recclnq 7404 . . . . . 6 ([⟨𝑗, 1o⟩] ~QQ → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
1513, 14syl 14 . . . . 5 (𝑗N → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
1615adantl 277 . . . 4 ((𝐴P𝑗N) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
17 simpl 109 . . . 4 ((𝐴P𝑗N) → 𝐴P)
18 nqprl 7563 . . . 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 411 . . 3 ((𝐴P𝑗N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st𝐴) ↔ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
2019rexbidva 2484 . 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 147 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 104  wb 105  wcel 2158  {cab 2173  wrex 2466  cop 3607   class class class wbr 4015  cfv 5228  1st c1st 6152  2nd c2nd 6153  1oc1o 6423  [cec 6546  Ncnpi 7284   ~Q ceq 7291  Qcnq 7292  *Qcrq 7296   <Q cltq 7297  Pcnp 7303  <P cltp 7307
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iltp 7482
This theorem is referenced by:  caucvgprprlemlim  7723
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