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Mirrors > Home > ILE Home > Th. List > halfnqq | Unicode version |
Description: One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.) |
Ref | Expression |
---|---|
halfnqq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nq 6464 | . . . . . . . . 9 | |
2 | addclnq 6473 | . . . . . . . . 9 | |
3 | 1, 1, 2 | mp2an 402 | . . . . . . . 8 |
4 | recclnq 6490 | . . . . . . . . 9 | |
5 | 3, 4 | ax-mp 7 | . . . . . . . 8 |
6 | distrnqg 6485 | . . . . . . . 8 | |
7 | 3, 5, 5, 6 | mp3an 1232 | . . . . . . 7 |
8 | recidnq 6491 | . . . . . . . . 9 | |
9 | 3, 8 | ax-mp 7 | . . . . . . . 8 |
10 | 9, 9 | oveq12i 5524 | . . . . . . 7 |
11 | 7, 10 | eqtri 2060 | . . . . . 6 |
12 | 11 | oveq1i 5522 | . . . . 5 |
13 | 9 | oveq2i 5523 | . . . . . 6 |
14 | addclnq 6473 | . . . . . . . . 9 | |
15 | 5, 5, 14 | mp2an 402 | . . . . . . . 8 |
16 | mulassnqg 6482 | . . . . . . . 8 | |
17 | 15, 3, 5, 16 | mp3an 1232 | . . . . . . 7 |
18 | mulcomnqg 6481 | . . . . . . . . 9 | |
19 | 15, 3, 18 | mp2an 402 | . . . . . . . 8 |
20 | 19 | oveq1i 5522 | . . . . . . 7 |
21 | 17, 20 | eqtr3i 2062 | . . . . . 6 |
22 | 4, 4, 14 | syl2anc 391 | . . . . . . 7 |
23 | mulidnq 6487 | . . . . . . 7 | |
24 | 3, 22, 23 | mp2b 8 | . . . . . 6 |
25 | 13, 21, 24 | 3eqtr3i 2068 | . . . . 5 |
26 | 12, 25, 9 | 3eqtr3i 2068 | . . . 4 |
27 | 26 | oveq2i 5523 | . . 3 |
28 | distrnqg 6485 | . . . 4 | |
29 | 5, 5, 28 | mp3an23 1224 | . . 3 |
30 | mulidnq 6487 | . . 3 | |
31 | 27, 29, 30 | 3eqtr3a 2096 | . 2 |
32 | mulclnq 6474 | . . . 4 | |
33 | 5, 32 | mpan2 401 | . . 3 |
34 | id 19 | . . . . . 6 | |
35 | 34, 34 | oveq12d 5530 | . . . . 5 |
36 | 35 | eqeq1d 2048 | . . . 4 |
37 | 36 | adantl 262 | . . 3 |
38 | 33, 37 | rspcedv 2660 | . 2 |
39 | 31, 38 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 wcel 1393 wrex 2307 cfv 4902 (class class class)co 5512 cnq 6378 c1q 6379 cplq 6380 cmq 6381 crq 6382 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 |
This theorem is referenced by: halfnq 6509 nsmallnqq 6510 subhalfnqq 6512 addlocpr 6634 addcanprleml 6712 addcanprlemu 6713 cauappcvgprlemm 6743 cauappcvgprlem1 6757 caucvgprlemm 6766 |
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