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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 7487 | . . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P) | |
2 | prml 7489 | . . . 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 7675 | . . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) | |
5 | 4 | ad2antrl 490 | . . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥) |
6 | 1 | ad2antrr 488 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P) |
7 | simplrr 536 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1st ‘𝐴)) | |
8 | prcdnql 7496 | . . . . . 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 411 | . . . . 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 2589 | . . . 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 2609 | . 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)) |
13 | nnnq 7434 | . . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q) | |
14 | recclnq 7404 | . . . . . 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 277 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q) |
17 | simpl 109 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P) | |
18 | nqprl 7563 | . . . 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 411 | . . 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 2484 | . 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 147 | 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 104 ↔ wb 105 ∈ wcel 2158 {cab 2173 ∃wrex 2466 ⟨cop 3607 class class class wbr 4015 ‘cfv 5228 1st c1st 6152 2nd c2nd 6153 1oc1o 6423 [cec 6546 Ncnpi 7284 ~Q ceq 7291 Qcnq 7292 *Qcrq 7296 <Q cltq 7297 Pcnp 7303 <P cltp 7307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-1o 6430 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-pli 7317 df-mi 7318 df-lti 7319 df-plpq 7356 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-plqqs 7361 df-mqqs 7362 df-1nqqs 7363 df-rq 7364 df-ltnqqs 7365 df-inp 7478 df-iltp 7482 |
This theorem is referenced by: caucvgprprlemlim 7723 |
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