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Mirrors > Home > ILE Home > Th. List > recexnq | Unicode version |
Description: Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.) |
Ref | Expression |
---|---|
recexnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6446 | . 2 | |
2 | oveq1 5519 | . . . . 5 | |
3 | 2 | eqeq1d 2048 | . . . 4 |
4 | 3 | anbi2d 437 | . . 3 |
5 | 4 | exbidv 1706 | . 2 |
6 | opelxpi 4376 | . . . . . 6 | |
7 | 6 | ancoms 255 | . . . . 5 |
8 | enqex 6458 | . . . . . 6 | |
9 | 8 | ecelqsi 6160 | . . . . 5 |
10 | 7, 9 | syl 14 | . . . 4 |
11 | 10, 1 | syl6eleqr 2131 | . . 3 |
12 | mulcompig 6429 | . . . . . . 7 | |
13 | 12 | opeq2d 3556 | . . . . . 6 |
14 | 13 | eceq1d 6142 | . . . . 5 |
15 | mulclpi 6426 | . . . . . 6 | |
16 | 1qec 6486 | . . . . . 6 | |
17 | 15, 16 | syl 14 | . . . . 5 |
18 | mulpipqqs 6471 | . . . . . . 7 | |
19 | 18 | an42s 523 | . . . . . 6 |
20 | 19 | anidms 377 | . . . . 5 |
21 | 14, 17, 20 | 3eqtr4rd 2083 | . . . 4 |
22 | 11, 21 | jca 290 | . . 3 |
23 | eleq1 2100 | . . . . 5 | |
24 | oveq2 5520 | . . . . . 6 | |
25 | 24 | eqeq1d 2048 | . . . . 5 |
26 | 23, 25 | anbi12d 442 | . . . 4 |
27 | 26 | spcegv 2641 | . . 3 |
28 | 11, 22, 27 | sylc 56 | . 2 |
29 | 1, 5, 28 | ecoptocl 6193 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cop 3378 cxp 4343 (class class class)co 5512 cec 6104 cqs 6105 cnpi 6370 cmi 6372 ceq 6377 cnq 6378 c1q 6379 cmq 6381 |
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-mi 6404 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-1nqqs 6449 |
This theorem is referenced by: recmulnqg 6489 recclnq 6490 |
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