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Theorem archrecpr 7874

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 7685	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7687	. . . 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 7873	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
5	4	ad2antrl 490	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
6	1	ad2antrr 488	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
7		simplrr 536	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1^st ‘𝐴))
8		prcdnql 7694	. . . . . 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 411	. . . . 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 2632	. . . 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 2653	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴))
13		nnnq 7632	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_o⟩] ~_Q ∈ Q)
14		recclnq 7602	. . . . . 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 7761	. . . 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 411	. . 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 2527	. 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 2200 {cab 2215 ∃wrex 2509 ⟨cop 3670 class class class wbr 4086 ‘cfv 5324 1^st c1st 6296 2^nd c2nd 6297 1_oc1o 6570 [cec 6695 Ncnpi 7482 ~_Q ceq 7489 Qcnq 7490 *_Qcrq 7494 <_Q cltq 7495 Pcnp 7501 <_P cltp 7505

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-eprel 4384 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-1o 6577 df-oadd 6581 df-omul 6582 df-er 6697 df-ec 6699 df-qs 6703 df-ni 7514 df-pli 7515 df-mi 7516 df-lti 7517 df-plpq 7554 df-mpq 7555 df-enq 7557 df-nqqs 7558 df-plqqs 7559 df-mqqs 7560 df-1nqqs 7561 df-rq 7562 df-ltnqqs 7563 df-inp 7676 df-iltp 7680

This theorem is referenced by: caucvgprprlemlim 7921

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