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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 2752 | . 2 ⊢ (*Q‘𝐴) = (*Q‘𝐴) | |
2 | recmulnq 9970 | . 2 ⊢ (𝐴 ∈ Q → ((*Q‘𝐴) = (*Q‘𝐴) ↔ (𝐴 ·Q (*Q‘𝐴)) = 1Q)) | |
3 | 1, 2 | mpbii 223 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ‘cfv 6041 (class class class)co 6805 Qcnq 9858 1Qc1q 9859 ·Q cmq 9862 *Qcrq 9863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-omul 7726 df-er 7903 df-ni 9878 df-mi 9880 df-lti 9881 df-mpq 9915 df-enq 9917 df-nq 9918 df-erq 9919 df-mq 9921 df-1nq 9922 df-rq 9923 |
This theorem is referenced by: recclnq 9972 recrecnq 9973 dmrecnq 9974 halfnq 9982 ltrnq 9985 addclprlem1 10022 addclprlem2 10023 mulclprlem 10025 1idpr 10035 prlem934 10039 prlem936 10053 reclem3pr 10055 reclem4pr 10056 |
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