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Mirrors > Home > ILE Home > Th. List > recrecnq | GIF version |
Description: Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) |
Ref | Expression |
---|---|
recrecnq | โข (๐ด โ Q โ (*Qโ(*Qโ๐ด)) = ๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclnq 7404 | . . . 4 โข (๐ด โ Q โ (*Qโ๐ด) โ Q) | |
2 | mulcomnqg 7395 | . . . 4 โข (((*Qโ๐ด) โ Q โง ๐ด โ Q) โ ((*Qโ๐ด) ยทQ ๐ด) = (๐ด ยทQ (*Qโ๐ด))) | |
3 | 1, 2 | mpancom 422 | . . 3 โข (๐ด โ Q โ ((*Qโ๐ด) ยทQ ๐ด) = (๐ด ยทQ (*Qโ๐ด))) |
4 | recidnq 7405 | . . 3 โข (๐ด โ Q โ (๐ด ยทQ (*Qโ๐ด)) = 1Q) | |
5 | 3, 4 | eqtrd 2220 | . 2 โข (๐ด โ Q โ ((*Qโ๐ด) ยทQ ๐ด) = 1Q) |
6 | recmulnqg 7403 | . . 3 โข (((*Qโ๐ด) โ Q โง ๐ด โ Q) โ ((*Qโ(*Qโ๐ด)) = ๐ด โ ((*Qโ๐ด) ยทQ ๐ด) = 1Q)) | |
7 | 1, 6 | mpancom 422 | . 2 โข (๐ด โ Q โ ((*Qโ(*Qโ๐ด)) = ๐ด โ ((*Qโ๐ด) ยทQ ๐ด) = 1Q)) |
8 | 5, 7 | mpbird 167 | 1 โข (๐ด โ Q โ (*Qโ(*Qโ๐ด)) = ๐ด) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 โ wb 105 = wceq 1363 โ wcel 2158 โcfv 5228 (class class class)co 5888 Qcnq 7292 1Qc1q 7293 ยทQ cmq 7295 *Qcrq 7296 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-1o 6430 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-mi 7318 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-mqqs 7362 df-1nqqs 7363 df-rq 7364 |
This theorem is referenced by: recexprlemm 7636 recexprlemloc 7643 archrecnq 7675 |
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