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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 7399 | . . 3 ⊢ (𝑁 ∈ N → [〈𝑁, 1o〉] ~Q ∈ Q) | |
2 | recclnq 7369 | . . . 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 7554 | . . 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 411 | . 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 7370 | . . . . . . 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 4012 | . . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) ↔ 𝑙 <Q 1Q)) |
9 | 8 | abbidv 2295 | . . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q}) |
10 | 7 | breq1d 4010 | . . . . 5 ⊢ (𝑁 ∈ N → (([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢)) |
11 | 10 | abbidv 2295 | . . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([〈𝑁, 1o〉] ~Q ·Q (*Q‘[〈𝑁, 1o〉] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}) |
12 | 9, 11 | opeq12d 3784 | . . 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 7444 | . . 3 ⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | |
14 | 12, 13 | eqtr4di 2228 | . 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 2212 | 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 1353 ∈ wcel 2148 {cab 2163 〈cop 3594 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 1oc1o 6403 [cec 6526 Ncnpi 7249 ~Q ceq 7256 Qcnq 7257 1Qc1q 7258 ·Q cmq 7260 *Qcrq 7261 <Q cltq 7262 1Pc1p 7269 ·P cmp 7271 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4285 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-1o 6410 df-2o 6411 df-oadd 6414 df-omul 6415 df-er 6528 df-ec 6530 df-qs 6534 df-ni 7281 df-pli 7282 df-mi 7283 df-lti 7284 df-plpq 7321 df-mpq 7322 df-enq 7324 df-nqqs 7325 df-plqqs 7326 df-mqqs 7327 df-1nqqs 7328 df-rq 7329 df-ltnqqs 7330 df-enq0 7401 df-nq0 7402 df-0nq0 7403 df-plq0 7404 df-mq0 7405 df-inp 7443 df-i1p 7444 df-imp 7446 |
This theorem is referenced by: recidpirq 7835 |
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