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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 7324 | . 2 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q) | |
2 | eqid 2164 | . . 3 ⊢ (*Q‘𝐴) = (*Q‘𝐴) | |
3 | recmulnqg 7323 | . . 3 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → ((*Q‘𝐴) = (*Q‘𝐴) ↔ (𝐴 ·Q (*Q‘𝐴)) = 1Q)) | |
4 | 2, 3 | mpbii 147 | . 2 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
5 | 1, 4 | mpdan 418 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ‘cfv 5182 (class class class)co 5836 Qcnq 7212 1Qc1q 7213 ·Q cmq 7215 *Qcrq 7216 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 |
This theorem is referenced by: recrecnq 7326 rec1nq 7327 halfnqq 7342 prarloclemarch 7350 ltrnqg 7352 addnqprllem 7459 addnqprulem 7460 addnqprl 7461 addnqpru 7462 appdivnq 7495 mulnqprl 7500 mulnqpru 7501 1idprl 7522 1idpru 7523 recexprlem1ssl 7565 recexprlem1ssu 7566 recexprlemss1l 7567 recexprlemss1u 7568 recidpipr 7788 |
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