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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 7407 | . . . 4 ⊢ (𝐴 ∈ P → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) | |
2 | prml 7409 | . . . 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 7595 | . . . . 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 7416 | . . . . . 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 2566 | . . . 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 2586 | . 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) ∈ (1st ‘𝐴)) |
13 | nnnq 7354 | . . . . . 6 ⊢ (𝑗 ∈ N → [〈𝑗, 1o〉] ~Q ∈ Q) | |
14 | recclnq 7324 | . . . . . 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 7483 | . . . 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 2461 | . 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 2135 {cab 2150 ∃wrex 2443 〈cop 3573 class class class wbr 3976 ‘cfv 5182 1st c1st 6098 2nd c2nd 6099 1oc1o 6368 [cec 6490 Ncnpi 7204 ~Q ceq 7211 Qcnq 7212 *Qcrq 7216 <Q cltq 7217 Pcnp 7223 <P cltp 7227 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-inp 7398 df-iltp 7402 |
This theorem is referenced by: caucvgprprlemlim 7643 |
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