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Theorem archrecpr 6762

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 6573	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 6575	. . . 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 6761	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑥)
5	4	ad2antrl 459	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑥)
6	1	ad2antrr 457	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
7		simplrr 488	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1^st ‘𝐴))
8		prcdnql 6582	. . . . . 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 391	. . . . 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 2421	. . . 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 2437	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ (1^st ‘𝐴))
13		nnnq 6520	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_𝑜⟩] ~_Q ∈ Q)
14		recclnq 6490	. . . . . 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 262	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q)
17		simpl 102	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 6649	. . . 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 391	. . 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 2323	. 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 135	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 97 ↔ wb 98 ∈ wcel 1393 {cab 2026 ∃wrex 2307 ⟨cop 3378 class class class wbr 3764 ‘cfv 4902 1^st c1st 5765 2^nd c2nd 5766 1_𝑜c1o 5994 [cec 6104 Ncnpi 6370 ~_Q ceq 6377 Qcnq 6378 *_Qcrq 6382 <_Q cltq 6383 Pcnp 6389 <_P cltp 6393

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-iltp 6568

This theorem is referenced by: caucvgprprlemlim 6809

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