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Mirrors > Home > ILE Home > Th. List > archrecpr | GIF version |
Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
Ref | Expression |
---|---|
archrecpr | ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7307 | . . . 4 ⊢ (𝐴 ∈ P → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) | |
2 | prml 7309 | . . . 4 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴)) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴)) |
4 | archrecnq 7495 | . . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑥) | |
5 | 4 | ad2antrl 482 | . . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑥) |
6 | 1 | ad2antrr 480 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) |
7 | simplrr 526 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1st ‘𝐴)) | |
8 | prcdnql 7316 | . . . . . 6 ⊢ ((〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ((*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑥 → (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 409 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ((*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑥 → (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴))) |
10 | 9 | reximdva 2537 | . . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → (∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑥 → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴))) |
11 | 5, 10 | mpd 13 | . . 3 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴)) |
12 | 3, 11 | rexlimddv 2557 | . 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴)) |
13 | nnnq 7254 | . . . . . 6 ⊢ (𝑗 ∈ N → [〈𝑗, 1o〉] ~Q ∈ Q) | |
14 | recclnq 7224 | . . . . . 6 ⊢ ([〈𝑗, 1o〉] ~Q ∈ Q → (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ Q) | |
15 | 13, 14 | syl 14 | . . . . 5 ⊢ (𝑗 ∈ N → (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ Q) |
16 | 15 | adantl 275 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ Q) |
17 | simpl 108 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P) | |
18 | nqprl 7383 | . . . 4 ⊢ (((*Q‘[〈𝑗, 1o〉] ~Q ) ∈ Q ∧ 𝐴 ∈ P) → ((*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴) ↔ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴)) | |
19 | 16, 17, 18 | syl2anc 409 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → ((*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴) ↔ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴)) |
20 | 19 | rexbidva 2435 | . 2 ⊢ (𝐴 ∈ P → (∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴) ↔ ∃𝑗 ∈ N 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴)) |
21 | 12, 20 | mpbid 146 | 1 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 {cab 2126 ∃wrex 2418 〈cop 3535 class class class wbr 3937 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 1oc1o 6314 [cec 6435 Ncnpi 7104 ~Q ceq 7111 Qcnq 7112 *Qcrq 7116 <Q cltq 7117 Pcnp 7123 <P cltp 7127 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-inp 7298 df-iltp 7302 |
This theorem is referenced by: caucvgprprlemlim 7543 |
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