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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 109 | . . . . 5 | |
2 | 0re 7878 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 1re 7877 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | 2, 4 | elicc2i 9843 | . . . . . . . 8 |
7 | 6 | simp1bi 997 | . . . . . . 7 |
8 | 7 | adantl 275 | . . . . . 6 |
9 | difrp 9599 | . . . . . . . 8 | |
10 | 9 | biimp3a 1327 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | eqid 2157 | . . . . . . 7 | |
13 | eqid 2157 | . . . . . . 7 | |
14 | 12, 13 | iccdil 9902 | . . . . . 6 |
15 | 3, 5, 8, 11, 14 | syl22anc 1221 | . . . . 5 |
16 | 1, 15 | mpbid 146 | . . . 4 |
17 | simpl2 986 | . . . . . . . 8 | |
18 | simpl1 985 | . . . . . . . 8 | |
19 | 17, 18 | resubcld 8256 | . . . . . . 7 |
20 | 19 | recnd 7906 | . . . . . 6 |
21 | 20 | mul02d 8267 | . . . . 5 |
22 | 20 | mulid2d 7896 | . . . . 5 |
23 | 21, 22 | oveq12d 5842 | . . . 4 |
24 | 16, 23 | eleqtrd 2236 | . . 3 |
25 | 8, 19 | remulcld 7908 | . . . 4 |
26 | eqid 2157 | . . . . 5 | |
27 | eqid 2157 | . . . . 5 | |
28 | 26, 27 | iccshftr 9898 | . . . 4 |
29 | 3, 19, 25, 18, 28 | syl22anc 1221 | . . 3 |
30 | 24, 29 | mpbid 146 | . 2 |
31 | 8 | recnd 7906 | . . . . 5 |
32 | 17 | recnd 7906 | . . . . 5 |
33 | 31, 32 | mulcld 7898 | . . . 4 |
34 | 18 | recnd 7906 | . . . . 5 |
35 | 31, 34 | mulcld 7898 | . . . 4 |
36 | 33, 35, 34 | subadd23d 8208 | . . 3 |
37 | 31, 32, 34 | subdid 8289 | . . . 4 |
38 | 37 | oveq1d 5839 | . . 3 |
39 | resubcl 8139 | . . . . . . . 8 | |
40 | 4, 8, 39 | sylancr 411 | . . . . . . 7 |
41 | 40, 18 | remulcld 7908 | . . . . . 6 |
42 | 41 | recnd 7906 | . . . . 5 |
43 | 42, 33 | addcomd 8026 | . . . 4 |
44 | 1cnd 7894 | . . . . . . 7 | |
45 | 44, 31, 34 | subdird 8290 | . . . . . 6 |
46 | 34 | mulid2d 7896 | . . . . . . 7 |
47 | 46 | oveq1d 5839 | . . . . . 6 |
48 | 45, 47 | eqtrd 2190 | . . . . 5 |
49 | 48 | oveq2d 5840 | . . . 4 |
50 | 43, 49 | eqtrd 2190 | . . 3 |
51 | 36, 38, 50 | 3eqtr4d 2200 | . 2 |
52 | 34 | addid2d 8025 | . . 3 |
53 | 32, 34 | npcand 8190 | . . 3 |
54 | 52, 53 | oveq12d 5842 | . 2 |
55 | 30, 51, 54 | 3eltr3d 2240 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wcel 2128 class class class wbr 3965 (class class class)co 5824 cr 7731 cc0 7732 c1 7733 caddc 7735 cmul 7737 clt 7912 cle 7913 cmin 8046 crp 9560 cicc 9795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-rp 9561 df-icc 9799 |
This theorem is referenced by: iccf1o 9908 |
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