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Theorem archrecpr 8025

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, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑗, 1_o⟩] ~_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 7836	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7838	. . . 4 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
3	1, 2	syl 14	. . 3 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
4		archrecnq 8024	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
5	4	ad2antrl 494	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
6	1	ad2antrr 492	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
7		simplrr 542	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1^st ‘𝐴))
8		prcdnql 7845	. . . . . 6 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → ((_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥 → (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴)))
9	6, 7, 8	syl2anc 415	. . . . 5 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ((_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥 → (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴)))
10	9	reximdva 2652	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → (∃𝑗 ∈ N (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥 → ∃𝑗 ∈ N (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴)))
11	5, 10	mpd 13	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴))
12	3, 11	rexlimddv 2673	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴))
13		nnnq 7783	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_o⟩] ~_Q ∈ Q)
14		recclnq 7753	. . . . . 6 ⊢ ([⟨𝑗, 1_o⟩] ~_Q ∈ Q → (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ Q)
15	13, 14	syl 14	. . . . 5 ⊢ (𝑗 ∈ N → (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ Q)
16	15	adantl 277	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ Q)
17		simpl 109	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 7912	. . . 4 ⊢ (((_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ Q ∧ 𝐴 ∈ P) → ((_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝐴))
19	16, 17, 18	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → ((_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )}, {𝑢 ∣ (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝐴))
20	19	rexbidva 2547	. 2 ⊢ (𝐴 ∈ P → (∃𝑗 ∈ N (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴) ↔ ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )}, {𝑢 ∣ (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝐴))
21	12, 20	mpbid 147	1 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑗, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2209 {cab 2224 ∃wrex 2529 ⟨cop 3711 class class class wbr 4128 ‘cfv 5375 1^st c1st 6366 2^nd c2nd 6367 1_oc1o 6674 [cec 6799 Ncnpi 7633 ~_Q ceq 7640 Qcnq 7641 *_Qcrq 7645 <_Q cltq 7646 Pcnp 7652 <_P cltp 7656

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-eprel 4432 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-1o 6681 df-oadd 6685 df-omul 6686 df-er 6801 df-ec 6803 df-qs 6807 df-ni 7665 df-pli 7666 df-mi 7667 df-lti 7668 df-plpq 7705 df-mpq 7706 df-enq 7708 df-nqqs 7709 df-plqqs 7710 df-mqqs 7711 df-1nqqs 7712 df-rq 7713 df-ltnqqs 7714 df-inp 7827 df-iltp 7831

This theorem is referenced by: caucvgprprlemlim 8072

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