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Mirrors > Home > ILE Home > Th. List > recidpirq | Unicode version |
Description: A real number times its reciprocal is one, where reciprocal is expressed with . (Contributed by Jim Kingdon, 15-Jul-2021.) |
Ref | Expression |
---|---|
recidpirq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnprlu 7361 | . . . 4 | |
2 | prsrcl 7592 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | recnnpr 7356 | . . . 4 | |
5 | prsrcl 7592 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | mulresr 7646 | . . 3 | |
8 | 3, 6, 7 | syl2anc 408 | . 2 |
9 | 1pr 7362 | . . . . . . . 8 | |
10 | 9 | a1i 9 | . . . . . . 7 |
11 | addclpr 7345 | . . . . . . 7 | |
12 | 1, 10, 11 | syl2anc 408 | . . . . . 6 |
13 | addclpr 7345 | . . . . . . 7 | |
14 | 4, 10, 13 | syl2anc 408 | . . . . . 6 |
15 | mulsrpr 7554 | . . . . . 6 | |
16 | 12, 10, 14, 10, 15 | syl22anc 1217 | . . . . 5 |
17 | recidpipr 7664 | . . . . . . 7 | |
18 | 1, 4, 17 | recidpirqlemcalc 7665 | . . . . . 6 |
19 | df-1r 7540 | . . . . . . . 8 | |
20 | 19 | eqeq2i 2150 | . . . . . . 7 |
21 | mulclpr 7380 | . . . . . . . . . 10 | |
22 | 12, 14, 21 | syl2anc 408 | . . . . . . . . 9 |
23 | 9, 9 | pm3.2i 270 | . . . . . . . . . 10 |
24 | mulclpr 7380 | . . . . . . . . . 10 | |
25 | 23, 24 | mp1i 10 | . . . . . . . . 9 |
26 | addclpr 7345 | . . . . . . . . 9 | |
27 | 22, 25, 26 | syl2anc 408 | . . . . . . . 8 |
28 | mulclpr 7380 | . . . . . . . . . 10 | |
29 | 12, 10, 28 | syl2anc 408 | . . . . . . . . 9 |
30 | mulclpr 7380 | . . . . . . . . . 10 | |
31 | 10, 14, 30 | syl2anc 408 | . . . . . . . . 9 |
32 | addclpr 7345 | . . . . . . . . 9 | |
33 | 29, 31, 32 | syl2anc 408 | . . . . . . . 8 |
34 | addclpr 7345 | . . . . . . . . 9 | |
35 | 23, 34 | mp1i 10 | . . . . . . . 8 |
36 | enreceq 7544 | . . . . . . . 8 | |
37 | 27, 33, 35, 10, 36 | syl22anc 1217 | . . . . . . 7 |
38 | 20, 37 | syl5bb 191 | . . . . . 6 |
39 | 18, 38 | mpbird 166 | . . . . 5 |
40 | 16, 39 | eqtrd 2172 | . . . 4 |
41 | 40 | opeq1d 3711 | . . 3 |
42 | df-1 7628 | . . 3 | |
43 | 41, 42 | syl6eqr 2190 | . 2 |
44 | 8, 43 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 cop 3530 class class class wbr 3929 cfv 5123 (class class class)co 5774 c1o 6306 cec 6427 cnpi 7080 ceq 7087 crq 7092 cltq 7093 cnp 7099 c1p 7100 cpp 7101 cmp 7102 cer 7104 cnr 7105 c0r 7106 c1r 7107 cmr 7110 c1 7621 cmul 7625 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-imp 7277 df-enr 7534 df-nr 7535 df-plr 7536 df-mr 7537 df-0r 7539 df-1r 7540 df-m1r 7541 df-c 7626 df-1 7628 df-mul 7632 |
This theorem is referenced by: recriota 7698 |
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