Theorem archrecpr 7812

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 7623	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prml 7625	. . . 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 7811	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
5	4	ad2antrl 490	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
6	1	ad2antrr 488	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		simplrr 536	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1st ‘𝐴))
8		prcdnql 7632	. . . . . 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 411	. . . . 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 2610	. . . 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 2630	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴))
13		nnnq 7570	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
14		recclnq 7540	. . . . . 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 277	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
17		simpl 109	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 7699	. . . 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 411	. . 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 2505	. 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 147	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 104   ↔ wb 105   ∈ wcel 2178  {cab 2193  ∃wrex 2487  ⟨cop 3646   class class class wbr 4059  ‘cfv 5290  1^st c1st 6247  2^nd c2nd 6248  1_oc1o 6518  [cec 6641  Ncnpi 7420   ~_Q ceq 7427  Qcnq 7428  *_Qcrq 7432   <_Q cltq 7433  Pcnp 7439  <_P cltp 7443

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-inp 7614  df-iltp 7618

This theorem is referenced by:  caucvgprprlemlim  7859