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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 7920 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 1re 7919 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | 2, 4 | elicc2i 9896 | . . . . . . . 8 |
7 | 6 | simp1bi 1007 | . . . . . . 7 |
8 | 7 | adantl 275 | . . . . . 6 |
9 | difrp 9649 | . . . . . . . 8 | |
10 | 9 | biimp3a 1340 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | eqid 2170 | . . . . . . 7 | |
13 | eqid 2170 | . . . . . . 7 | |
14 | 12, 13 | iccdil 9955 | . . . . . 6 |
15 | 3, 5, 8, 11, 14 | syl22anc 1234 | . . . . 5 |
16 | 1, 15 | mpbid 146 | . . . 4 |
17 | simpl2 996 | . . . . . . . 8 | |
18 | simpl1 995 | . . . . . . . 8 | |
19 | 17, 18 | resubcld 8300 | . . . . . . 7 |
20 | 19 | recnd 7948 | . . . . . 6 |
21 | 20 | mul02d 8311 | . . . . 5 |
22 | 20 | mulid2d 7938 | . . . . 5 |
23 | 21, 22 | oveq12d 5871 | . . . 4 |
24 | 16, 23 | eleqtrd 2249 | . . 3 |
25 | 8, 19 | remulcld 7950 | . . . 4 |
26 | eqid 2170 | . . . . 5 | |
27 | eqid 2170 | . . . . 5 | |
28 | 26, 27 | iccshftr 9951 | . . . 4 |
29 | 3, 19, 25, 18, 28 | syl22anc 1234 | . . 3 |
30 | 24, 29 | mpbid 146 | . 2 |
31 | 8 | recnd 7948 | . . . . 5 |
32 | 17 | recnd 7948 | . . . . 5 |
33 | 31, 32 | mulcld 7940 | . . . 4 |
34 | 18 | recnd 7948 | . . . . 5 |
35 | 31, 34 | mulcld 7940 | . . . 4 |
36 | 33, 35, 34 | subadd23d 8252 | . . 3 |
37 | 31, 32, 34 | subdid 8333 | . . . 4 |
38 | 37 | oveq1d 5868 | . . 3 |
39 | resubcl 8183 | . . . . . . . 8 | |
40 | 4, 8, 39 | sylancr 412 | . . . . . . 7 |
41 | 40, 18 | remulcld 7950 | . . . . . 6 |
42 | 41 | recnd 7948 | . . . . 5 |
43 | 42, 33 | addcomd 8070 | . . . 4 |
44 | 1cnd 7936 | . . . . . . 7 | |
45 | 44, 31, 34 | subdird 8334 | . . . . . 6 |
46 | 34 | mulid2d 7938 | . . . . . . 7 |
47 | 46 | oveq1d 5868 | . . . . . 6 |
48 | 45, 47 | eqtrd 2203 | . . . . 5 |
49 | 48 | oveq2d 5869 | . . . 4 |
50 | 43, 49 | eqtrd 2203 | . . 3 |
51 | 36, 38, 50 | 3eqtr4d 2213 | . 2 |
52 | 34 | addid2d 8069 | . . 3 |
53 | 32, 34 | npcand 8234 | . . 3 |
54 | 52, 53 | oveq12d 5871 | . 2 |
55 | 30, 51, 54 | 3eltr3d 2253 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wcel 2141 class class class wbr 3989 (class class class)co 5853 cr 7773 cc0 7774 c1 7775 caddc 7777 cmul 7779 clt 7954 cle 7955 cmin 8090 crp 9610 cicc 9848 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-rp 9611 df-icc 9852 |
This theorem is referenced by: iccf1o 9961 |
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