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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 7766 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 1re 7765 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | 2, 4 | elicc2i 9722 | . . . . . . . 8 |
7 | 6 | simp1bi 996 | . . . . . . 7 |
8 | 7 | adantl 275 | . . . . . 6 |
9 | difrp 9480 | . . . . . . . 8 | |
10 | 9 | biimp3a 1323 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | eqid 2139 | . . . . . . 7 | |
13 | eqid 2139 | . . . . . . 7 | |
14 | 12, 13 | iccdil 9781 | . . . . . 6 |
15 | 3, 5, 8, 11, 14 | syl22anc 1217 | . . . . 5 |
16 | 1, 15 | mpbid 146 | . . . 4 |
17 | simpl2 985 | . . . . . . . 8 | |
18 | simpl1 984 | . . . . . . . 8 | |
19 | 17, 18 | resubcld 8143 | . . . . . . 7 |
20 | 19 | recnd 7794 | . . . . . 6 |
21 | 20 | mul02d 8154 | . . . . 5 |
22 | 20 | mulid2d 7784 | . . . . 5 |
23 | 21, 22 | oveq12d 5792 | . . . 4 |
24 | 16, 23 | eleqtrd 2218 | . . 3 |
25 | 8, 19 | remulcld 7796 | . . . 4 |
26 | eqid 2139 | . . . . 5 | |
27 | eqid 2139 | . . . . 5 | |
28 | 26, 27 | iccshftr 9777 | . . . 4 |
29 | 3, 19, 25, 18, 28 | syl22anc 1217 | . . 3 |
30 | 24, 29 | mpbid 146 | . 2 |
31 | 8 | recnd 7794 | . . . . 5 |
32 | 17 | recnd 7794 | . . . . 5 |
33 | 31, 32 | mulcld 7786 | . . . 4 |
34 | 18 | recnd 7794 | . . . . 5 |
35 | 31, 34 | mulcld 7786 | . . . 4 |
36 | 33, 35, 34 | subadd23d 8095 | . . 3 |
37 | 31, 32, 34 | subdid 8176 | . . . 4 |
38 | 37 | oveq1d 5789 | . . 3 |
39 | resubcl 8026 | . . . . . . . 8 | |
40 | 4, 8, 39 | sylancr 410 | . . . . . . 7 |
41 | 40, 18 | remulcld 7796 | . . . . . 6 |
42 | 41 | recnd 7794 | . . . . 5 |
43 | 42, 33 | addcomd 7913 | . . . 4 |
44 | 1cnd 7782 | . . . . . . 7 | |
45 | 44, 31, 34 | subdird 8177 | . . . . . 6 |
46 | 34 | mulid2d 7784 | . . . . . . 7 |
47 | 46 | oveq1d 5789 | . . . . . 6 |
48 | 45, 47 | eqtrd 2172 | . . . . 5 |
49 | 48 | oveq2d 5790 | . . . 4 |
50 | 43, 49 | eqtrd 2172 | . . 3 |
51 | 36, 38, 50 | 3eqtr4d 2182 | . 2 |
52 | 34 | addid2d 7912 | . . 3 |
53 | 32, 34 | npcand 8077 | . . 3 |
54 | 52, 53 | oveq12d 5792 | . 2 |
55 | 30, 51, 54 | 3eltr3d 2222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cle 7801 cmin 7933 crp 9441 cicc 9674 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-rp 9442 df-icc 9678 |
This theorem is referenced by: iccf1o 9787 |
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