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Mirrors > Home > ILE Home > Th. List > iccdil | Unicode version |
Description: Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
iccdil.1 | |
iccdil.2 |
Ref | Expression |
---|---|
iccdil |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . . 5 | |
2 | rpre 8589 | . . . . . 6 | |
3 | remulcl 7009 | . . . . . 6 | |
4 | 2, 3 | sylan2 270 | . . . . 5 |
5 | 1, 4 | 2thd 164 | . . . 4 |
6 | 5 | adantl 262 | . . 3 |
7 | elrp 8585 | . . . . . . 7 | |
8 | lemul1 7584 | . . . . . . 7 | |
9 | 7, 8 | syl3an3b 1173 | . . . . . 6 |
10 | 9 | 3expb 1105 | . . . . 5 |
11 | 10 | adantlr 446 | . . . 4 |
12 | iccdil.1 | . . . . 5 | |
13 | 12 | breq1i 3771 | . . . 4 |
14 | 11, 13 | syl6bb 185 | . . 3 |
15 | lemul1 7584 | . . . . . . . 8 | |
16 | 7, 15 | syl3an3b 1173 | . . . . . . 7 |
17 | 16 | 3expb 1105 | . . . . . 6 |
18 | 17 | an12s 499 | . . . . 5 |
19 | 18 | adantll 445 | . . . 4 |
20 | iccdil.2 | . . . . 5 | |
21 | 20 | breq2i 3772 | . . . 4 |
22 | 19, 21 | syl6bb 185 | . . 3 |
23 | 6, 14, 22 | 3anbi123d 1207 | . 2 |
24 | elicc2 8807 | . . 3 | |
25 | 24 | adantr 261 | . 2 |
26 | remulcl 7009 | . . . . . . 7 | |
27 | 12, 26 | syl5eqelr 2125 | . . . . . 6 |
28 | remulcl 7009 | . . . . . . 7 | |
29 | 20, 28 | syl5eqelr 2125 | . . . . . 6 |
30 | elicc2 8807 | . . . . . 6 | |
31 | 27, 29, 30 | syl2an 273 | . . . . 5 |
32 | 31 | anandirs 527 | . . . 4 |
33 | 2, 32 | sylan2 270 | . . 3 |
34 | 33 | adantrl 447 | . 2 |
35 | 23, 25, 34 | 3bitr4d 209 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cc0 6889 cmul 6894 clt 7060 cle 7061 crp 8583 cicc 8760 |
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-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 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-ltadd 7000 ax-pre-mulgt0 7001 |
This theorem depends on definitions: df-bi 110 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-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-po 4033 df-iso 4034 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-rp 8584 df-icc 8764 |
This theorem is referenced by: iccdili 8867 lincmb01cmp 8871 iccf1o 8872 |
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