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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 7410 | . . . 4 โข (๐ด โ Q โ (*Qโ๐ด) โ Q) | |
2 | mulcomnqg 7401 | . . . 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 7411 | . . 3 โข (๐ด โ Q โ (๐ด ยทQ (*Qโ๐ด)) = 1Q) | |
5 | 3, 4 | eqtrd 2222 | . 2 โข (๐ด โ Q โ ((*Qโ๐ด) ยทQ ๐ด) = 1Q) |
6 | recmulnqg 7409 | . . 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 1364 โ wcel 2160 โcfv 5231 (class class class)co 5891 Qcnq 7298 1Qc1q 7299 ยทQ cmq 7301 *Qcrq 7302 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-1o 6435 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7322 df-mi 7324 df-mpq 7363 df-enq 7365 df-nqqs 7366 df-mqqs 7368 df-1nqqs 7369 df-rq 7370 |
This theorem is referenced by: recexprlemm 7642 recexprlemloc 7649 archrecnq 7681 |
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