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Theorem archrecpr 7626

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 7437	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7439	. . . 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 7625	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
5	4	ad2antrl 487	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) <_Q 𝑥)
6	1	ad2antrr 485	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
7		simplrr 531	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1^st ‘𝐴))
8		prcdnql 7446	. . . . . 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 409	. . . . 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 2572	. . . 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 2592	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ (1^st ‘𝐴))
13		nnnq 7384	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1_o⟩] ~_Q ∈ Q)
14		recclnq 7354	. . . . . 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 275	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*_Q‘[⟨𝑗, 1_o⟩] ~_Q ) ∈ Q)
17		simpl 108	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 7513	. . . 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 409	. . 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 2467	. 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 146	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 103 ↔ wb 104 ∈ wcel 2141 {cab 2156 ∃wrex 2449 ⟨cop 3586 class class class wbr 3989 ‘cfv 5198 1^st c1st 6117 2^nd c2nd 6118 1_oc1o 6388 [cec 6511 Ncnpi 7234 ~_Q ceq 7241 Qcnq 7242 *_Qcrq 7246 <_Q cltq 7247 Pcnp 7253 <_P cltp 7257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-inp 7428 df-iltp 7432

This theorem is referenced by: caucvgprprlemlim 7673

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