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Mirrors > Home > ILE Home > Th. List > recidpipr | GIF version |
Description: Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.) |
Ref | Expression |
---|---|
recidpipr | ⊢ (𝑁 ∈ N → (〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 ·P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉) = 1P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnq 7230 | . . 3 ⊢ (𝑁 ∈ N → [〈𝑁, 1o〉] ~Q ∈ Q) | |
2 | recclnq 7200 | . . . 4 ⊢ ([〈𝑁, 1o〉] ~Q ∈ Q → (*Q‘[〈𝑁, 1o〉] ~Q ) ∈ Q) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝑁 ∈ N → (*Q‘[〈𝑁, 1o〉] ~Q ) ∈ Q) |
4 | mulnqpr 7385 | . . 3 ⊢ (([〈𝑁, 1o〉] ~Q ∈ Q ∧ (*Q‘[〈𝑁, 1o〉] ~Q ) ∈ Q) → 〈{𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))}, {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 ·P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉)) | |
5 | 1, 3, 4 | syl2anc 408 | . 2 ⊢ (𝑁 ∈ N → 〈{𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))}, {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢}〉 = (〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 ·P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉)) |
6 | recidnq 7201 | . . . . . . 7 ⊢ ([〈𝑁, 1o〉] ~Q ∈ Q → ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) = 1Q) | |
7 | 1, 6 | syl 14 | . . . . . 6 ⊢ (𝑁 ∈ N → ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) = 1Q) |
8 | 7 | breq2d 3941 | . . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) ↔ 𝑙 <Q 1Q)) |
9 | 8 | abbidv 2257 | . . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q}) |
10 | 7 | breq1d 3939 | . . . . 5 ⊢ (𝑁 ∈ N → (([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢)) |
11 | 10 | abbidv 2257 | . . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}) |
12 | 9, 11 | opeq12d 3713 | . . 3 ⊢ (𝑁 ∈ N → 〈{𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))}, {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢}〉 = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉) |
13 | df-i1p 7275 | . . 3 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
14 | 12, 13 | syl6eqr 2190 | . 2 ⊢ (𝑁 ∈ N → 〈{𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))}, {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢}〉 = 1P) |
15 | 5, 14 | eqtr3d 2174 | 1 ⊢ (𝑁 ∈ N → (〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 ·P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉) = 1P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 {cab 2125 〈cop 3530 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1oc1o 6306 [cec 6427 Ncnpi 7080 ~Q ceq 7087 Qcnq 7088 1Qc1q 7089 ·Q cmq 7091 *Qcrq 7092 <Q cltq 7093 1Pc1p 7100 ·P cmp 7102 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-imp 7277 |
This theorem is referenced by: recidpirq 7666 |
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