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Theorem archrecpr 6854

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 6665	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 6667	. . . 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 6853	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑥)
5	4	ad2antrl 473	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑥)
6	1	ad2antrr 471	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
7		simplrr 502	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1^st ‘𝐴))
8		prcdnql 6674	. . . . . 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 403	. . . . 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 2463	. . . 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 2481	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ (1^st ‘𝐴))
13		nnnq 6612	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_𝑜⟩] ~_Q ∈ Q)
14		recclnq 6582	. . . . . 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 271	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) ∈ Q)
17		simpl 107	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 6741	. . . 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 403	. . 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 2365	. 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 145	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 102 ↔ wb 103 ∈ wcel 1433 {cab 2067 ∃wrex 2349 ⟨cop 3401 class class class wbr 3785 ‘cfv 4922 1^st c1st 5785 2^nd c2nd 5786 1_𝑜c1o 6017 [cec 6127 Ncnpi 6462 ~_Q ceq 6469 Qcnq 6470 *_Qcrq 6474 <_Q cltq 6475 Pcnp 6481 <_P cltp 6485

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-inp 6656 df-iltp 6660

This theorem is referenced by: caucvgprprlemlim 6901

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