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Mirrors > Home > ILE Home > Th. List > archrecnq | GIF version |
Description: Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
Ref | Expression |
---|---|
archrecnq | ⊢ (𝐴 ∈ Q → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclnq 7324 | . . 3 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q) | |
2 | archnqq 7349 | . . 3 ⊢ ((*Q‘𝐴) ∈ Q → ∃𝑗 ∈ N (*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑗 ∈ N (*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ) |
4 | nnnq 7354 | . . . . 5 ⊢ (𝑗 ∈ N → [〈𝑗, 1o〉] ~Q ∈ Q) | |
5 | ltrnqg 7352 | . . . . 5 ⊢ (((*Q‘𝐴) ∈ Q ∧ [〈𝑗, 1o〉] ~Q ∈ Q) → ((*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ↔ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q (*Q‘(*Q‘𝐴)))) | |
6 | 1, 4, 5 | syl2an 287 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝑗 ∈ N) → ((*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ↔ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q (*Q‘(*Q‘𝐴)))) |
7 | recrecnq 7326 | . . . . . 6 ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴) | |
8 | 7 | breq2d 3988 | . . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘[〈𝑗, 1o〉] ~Q ) <Q (*Q‘(*Q‘𝐴)) ↔ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴)) |
9 | 8 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝑗 ∈ N) → ((*Q‘[〈𝑗, 1o〉] ~Q ) <Q (*Q‘(*Q‘𝐴)) ↔ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴)) |
10 | 6, 9 | bitrd 187 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝑗 ∈ N) → ((*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ↔ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴)) |
11 | 10 | rexbidva 2461 | . 2 ⊢ (𝐴 ∈ Q → (∃𝑗 ∈ N (*Q‘𝐴) <Q [〈𝑗, 1o〉] ~Q ↔ ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴)) |
12 | 3, 11 | mpbid 146 | 1 ⊢ (𝐴 ∈ Q → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 ∃wrex 2443 〈cop 3573 class class class wbr 3976 ‘cfv 5182 1oc1o 6368 [cec 6490 Ncnpi 7204 ~Q ceq 7211 Qcnq 7212 *Qcrq 7216 <Q cltq 7217 |
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-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-mpq 7277 df-enq 7279 df-nqqs 7280 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 |
This theorem is referenced by: archrecpr 7596 caucvgprlemm 7600 caucvgprlemloc 7607 caucvgprlemlim 7613 caucvgprprlemml 7626 caucvgprprlemloc 7635 |
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