Theorem archrecpr 7472

Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)

Assertion

Ref	Expression
archrecpr	⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <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.

Step	Hyp	Ref	Expression
1		prop 7283	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prml 7285	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
3	1, 2	syl 14	. . 3 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
4		archrecnq 7471	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
5	4	ad2antrl 481	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
6	1	ad2antrr 479	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		simplrr 525	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1st ‘𝐴))
8		prcdnql 7292	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
9	6, 7, 8	syl2anc 408	. . . . 5 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
10	9	reximdva 2534	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
11	5, 10	mpd 13	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴))
12	3, 11	rexlimddv 2554	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴))
13		nnnq 7230	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
14		recclnq 7200	. . . . . 6 ⊢ ([⟨𝑗, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
15	13, 14	syl 14	. . . . 5 ⊢ (𝑗 ∈ N → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
16	15	adantl 275	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
17		simpl 108	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 7359	. . . 4 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝐴 ∈ P) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
19	16, 17, 18	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
20	19	rexbidva 2434	. 2 ⊢ (𝐴 ∈ P → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
21	12, 20	mpbid 146	1 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  {cab 2125  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  1^st c1st 6036  2^nd c2nd 6037  1_oc1o 6306  [cec 6427  Ncnpi 7080   ~_Q ceq 7087  Qcnq 7088  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-iltp 7278

This theorem is referenced by:  caucvgprprlemlim  7519