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Mirrors > Home > ILE Home > Th. List > abslt | Unicode version |
Description: Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abslt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 495 | . . . . . 6 | |
2 | 1 | renegcld 7484 | . . . . 5 |
3 | 1 | recnd 7147 | . . . . . 6 |
4 | abscl 9937 | . . . . . 6 | |
5 | 3, 4 | syl 14 | . . . . 5 |
6 | simplr 496 | . . . . 5 | |
7 | leabs 9960 | . . . . . . 7 | |
8 | 2, 7 | syl 14 | . . . . . 6 |
9 | absneg 9936 | . . . . . . 7 | |
10 | 3, 9 | syl 14 | . . . . . 6 |
11 | 8, 10 | breqtrd 3809 | . . . . 5 |
12 | simpr 108 | . . . . 5 | |
13 | 2, 5, 6, 11, 12 | lelttrd 7234 | . . . 4 |
14 | leabs 9960 | . . . . . 6 | |
15 | 14 | ad2antrr 471 | . . . . 5 |
16 | 1, 5, 6, 15, 12 | lelttrd 7234 | . . . 4 |
17 | 13, 16 | jca 300 | . . 3 |
18 | simpll 495 | . . . . . 6 | |
19 | simpl 107 | . . . . . . . . 9 | |
20 | 19 | recnd 7147 | . . . . . . . 8 |
21 | 20, 9 | syl 14 | . . . . . . 7 |
22 | 19 | renegcld 7484 | . . . . . . . 8 |
23 | 0red 7120 | . . . . . . . . . 10 | |
24 | ltabs 9973 | . . . . . . . . . 10 | |
25 | 19, 23, 24 | ltled 7228 | . . . . . . . . 9 |
26 | 19 | le0neg1d 7618 | . . . . . . . . 9 |
27 | 25, 26 | mpbid 145 | . . . . . . . 8 |
28 | absid 9957 | . . . . . . . 8 | |
29 | 22, 27, 28 | syl2anc 403 | . . . . . . 7 |
30 | 21, 29 | eqtr3d 2115 | . . . . . 6 |
31 | 18, 30 | sylan 277 | . . . . 5 |
32 | simplrl 501 | . . . . 5 | |
33 | 31, 32 | eqbrtrd 3805 | . . . 4 |
34 | simpr 108 | . . . 4 | |
35 | simprr 498 | . . . . 5 | |
36 | simplr 496 | . . . . . 6 | |
37 | 18 | recnd 7147 | . . . . . . 7 |
38 | 37, 4 | syl 14 | . . . . . 6 |
39 | axltwlin 7180 | . . . . . 6 | |
40 | 18, 36, 38, 39 | syl3anc 1169 | . . . . 5 |
41 | 35, 40 | mpd 13 | . . . 4 |
42 | 33, 34, 41 | mpjaodan 744 | . . 3 |
43 | 17, 42 | impbida 560 | . 2 |
44 | ltnegcon1 7567 | . . 3 | |
45 | 44 | anbi1d 452 | . 2 |
46 | 43, 45 | bitrd 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 class class class wbr 3785 cfv 4922 cc 6979 cr 6980 cc0 6981 clt 7153 cle 7154 cneg 7280 cabs 9883 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 |
This theorem is referenced by: absdiflt 9978 abslti 10024 absltd 10060 |
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