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Mirrors > Home > ILE Home > Th. List > abs00ap | Unicode version |
Description: The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.) |
Ref | Expression |
---|---|
abs00ap | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absval2 9655 | . . . . . . . . . 10 | |
2 | 1 | breq1d 3774 | . . . . . . . . 9 # # |
3 | sqrt0 9602 | . . . . . . . . . 10 | |
4 | 3 | breq2i 3772 | . . . . . . . . 9 # # |
5 | 2, 4 | syl6bbr 187 | . . . . . . . 8 # # |
6 | recl 9453 | . . . . . . . . . . 11 | |
7 | 6 | resqcld 9406 | . . . . . . . . . 10 |
8 | imcl 9454 | . . . . . . . . . . 11 | |
9 | 8 | resqcld 9406 | . . . . . . . . . 10 |
10 | 7, 9 | readdcld 7055 | . . . . . . . . 9 |
11 | 6 | sqge0d 9407 | . . . . . . . . . 10 |
12 | 8 | sqge0d 9407 | . . . . . . . . . 10 |
13 | 7, 9, 11, 12 | addge0d 7513 | . . . . . . . . 9 |
14 | 0red 7028 | . . . . . . . . 9 | |
15 | 14 | leidd 7506 | . . . . . . . . 9 |
16 | sqrt11ap 9636 | . . . . . . . . 9 # # | |
17 | 10, 13, 14, 15, 16 | syl22anc 1136 | . . . . . . . 8 # # |
18 | 5, 17 | bitrd 177 | . . . . . . 7 # # |
19 | 00id 7154 | . . . . . . . 8 | |
20 | 19 | breq2i 3772 | . . . . . . 7 # # |
21 | 18, 20 | syl6bbr 187 | . . . . . 6 # # |
22 | 7 | recnd 7054 | . . . . . . 7 |
23 | 9 | recnd 7054 | . . . . . . 7 |
24 | 0cnd 7020 | . . . . . . 7 | |
25 | addext 7601 | . . . . . . 7 # # # | |
26 | 22, 23, 24, 24, 25 | syl22anc 1136 | . . . . . 6 # # # |
27 | 21, 26 | sylbid 139 | . . . . 5 # # # |
28 | 6 | recnd 7054 | . . . . . . 7 |
29 | 2nn 8077 | . . . . . . 7 | |
30 | expap0 9285 | . . . . . . 7 # # | |
31 | 28, 29, 30 | sylancl 392 | . . . . . 6 # # |
32 | 8 | recnd 7054 | . . . . . . 7 |
33 | expap0 9285 | . . . . . . 7 # # | |
34 | 32, 29, 33 | sylancl 392 | . . . . . 6 # # |
35 | 31, 34 | orbi12d 707 | . . . . 5 # # # # |
36 | 27, 35 | sylibd 138 | . . . 4 # # # |
37 | crap0 7910 | . . . . 5 # # # | |
38 | 6, 8, 37 | syl2anc 391 | . . . 4 # # # |
39 | 36, 38 | sylibd 138 | . . 3 # # |
40 | replim 9459 | . . . 4 | |
41 | 40 | breq1d 3774 | . . 3 # # |
42 | 39, 41 | sylibrd 158 | . 2 # # |
43 | absrpclap 9659 | . . . 4 # | |
44 | 43 | rpap0d 8628 | . . 3 # # |
45 | 44 | ex 108 | . 2 # # |
46 | 42, 45 | impbid 120 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wcel 1393 class class class wbr 3764 cfv 4902 (class class class)co 5512 cc 6887 cr 6888 cc0 6889 ci 6891 caddc 6892 cmul 6894 cle 7061 # cap 7572 cn 7914 c2 7964 cexp 9254 cre 9440 cim 9441 csqrt 9594 cabs 9595 |
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 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 ax-caucvg 7004 |
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-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 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-if 3332 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-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 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-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 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-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255 df-cj 9442 df-re 9443 df-im 9444 df-rsqrt 9596 df-abs 9597 |
This theorem is referenced by: abs00 9662 absexpzap 9676 ltabs 9683 recvalap 9693 absgt0ap 9695 |
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