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Mirrors > Home > ILE Home > Th. List > lincmb01cmp | Unicode version |
Description: A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.) |
Ref | Expression |
---|---|
lincmb01cmp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . . . 5 | |
2 | 0re 7027 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 1re 7026 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | 2, 4 | elicc2i 8808 | . . . . . . . 8 |
7 | 6 | simp1bi 919 | . . . . . . 7 |
8 | 7 | adantl 262 | . . . . . 6 |
9 | difrp 8619 | . . . . . . . 8 | |
10 | 9 | biimp3a 1235 | . . . . . . 7 |
11 | 10 | adantr 261 | . . . . . 6 |
12 | eqid 2040 | . . . . . . 7 | |
13 | eqid 2040 | . . . . . . 7 | |
14 | 12, 13 | iccdil 8866 | . . . . . 6 |
15 | 3, 5, 8, 11, 14 | syl22anc 1136 | . . . . 5 |
16 | 1, 15 | mpbid 135 | . . . 4 |
17 | simpl2 908 | . . . . . . . 8 | |
18 | simpl1 907 | . . . . . . . 8 | |
19 | 17, 18 | resubcld 7379 | . . . . . . 7 |
20 | 19 | recnd 7054 | . . . . . 6 |
21 | 20 | mul02d 7389 | . . . . 5 |
22 | 20 | mulid2d 7045 | . . . . 5 |
23 | 21, 22 | oveq12d 5530 | . . . 4 |
24 | 16, 23 | eleqtrd 2116 | . . 3 |
25 | 8, 19 | remulcld 7056 | . . . 4 |
26 | eqid 2040 | . . . . 5 | |
27 | eqid 2040 | . . . . 5 | |
28 | 26, 27 | iccshftr 8862 | . . . 4 |
29 | 3, 19, 25, 18, 28 | syl22anc 1136 | . . 3 |
30 | 24, 29 | mpbid 135 | . 2 |
31 | 8 | recnd 7054 | . . . . 5 |
32 | 17 | recnd 7054 | . . . . 5 |
33 | 31, 32 | mulcld 7047 | . . . 4 |
34 | 18 | recnd 7054 | . . . . 5 |
35 | 31, 34 | mulcld 7047 | . . . 4 |
36 | 33, 35, 34 | subadd23d 7344 | . . 3 |
37 | 31, 32, 34 | subdid 7411 | . . . 4 |
38 | 37 | oveq1d 5527 | . . 3 |
39 | resubcl 7275 | . . . . . . . 8 | |
40 | 4, 8, 39 | sylancr 393 | . . . . . . 7 |
41 | 40, 18 | remulcld 7056 | . . . . . 6 |
42 | 41 | recnd 7054 | . . . . 5 |
43 | 42, 33 | addcomd 7164 | . . . 4 |
44 | 1cnd 7043 | . . . . . . 7 | |
45 | 44, 31, 34 | subdird 7412 | . . . . . 6 |
46 | 34 | mulid2d 7045 | . . . . . . 7 |
47 | 46 | oveq1d 5527 | . . . . . 6 |
48 | 45, 47 | eqtrd 2072 | . . . . 5 |
49 | 48 | oveq2d 5528 | . . . 4 |
50 | 43, 49 | eqtrd 2072 | . . 3 |
51 | 36, 38, 50 | 3eqtr4d 2082 | . 2 |
52 | 34 | addid2d 7163 | . . 3 |
53 | 32, 34 | npcand 7326 | . . 3 |
54 | 52, 53 | oveq12d 5530 | . 2 |
55 | 30, 51, 54 | 3eltr3d 2120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cc0 6889 c1 6890 caddc 6892 cmul 6894 clt 7060 cle 7061 cmin 7182 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: iccf1o 8872 |
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