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Mirrors > Home > MPE Home > Th. List > recidnq | Structured version Visualization version GIF version |
Description: A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
recidnq | ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ (*Q‘𝐴) = (*Q‘𝐴) | |
2 | recmulnq 10380 | . 2 ⊢ (𝐴 ∈ Q → ((*Q‘𝐴) = (*Q‘𝐴) ↔ (𝐴 ·Q (*Q‘𝐴)) = 1Q)) | |
3 | 1, 2 | mpbii 235 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Qcnq 10268 1Qc1q 10269 ·Q cmq 10272 *Qcrq 10273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-mi 10290 df-lti 10291 df-mpq 10325 df-enq 10327 df-nq 10328 df-erq 10329 df-mq 10331 df-1nq 10332 df-rq 10333 |
This theorem is referenced by: recclnq 10382 recrecnq 10383 dmrecnq 10384 halfnq 10392 ltrnq 10395 addclprlem1 10432 addclprlem2 10433 mulclprlem 10435 1idpr 10445 prlem934 10449 prlem936 10463 reclem3pr 10465 reclem4pr 10466 |
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