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Mirrors > Home > ILE Home > Th. List > recidnq | GIF version |
Description: A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) |
Ref | Expression |
---|---|
recidnq | ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclnq 7390 | . 2 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q) | |
2 | eqid 2177 | . . 3 ⊢ (*Q‘𝐴) = (*Q‘𝐴) | |
3 | recmulnqg 7389 | . . 3 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → ((*Q‘𝐴) = (*Q‘𝐴) ↔ (𝐴 ·Q (*Q‘𝐴)) = 1Q)) | |
4 | 2, 3 | mpbii 148 | . 2 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
5 | 1, 4 | mpdan 421 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Qcnq 7278 1Qc1q 7279 ·Q cmq 7281 *Qcrq 7282 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-mi 7304 df-mpq 7343 df-enq 7345 df-nqqs 7346 df-mqqs 7348 df-1nqqs 7349 df-rq 7350 |
This theorem is referenced by: recrecnq 7392 rec1nq 7393 halfnqq 7408 prarloclemarch 7416 ltrnqg 7418 addnqprllem 7525 addnqprulem 7526 addnqprl 7527 addnqpru 7528 appdivnq 7561 mulnqprl 7566 mulnqpru 7567 1idprl 7588 1idpru 7589 recexprlem1ssl 7631 recexprlem1ssu 7632 recexprlemss1l 7633 recexprlemss1u 7634 recidpipr 7854 |
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