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Mirrors > Home > ILE Home > Th. List > halfpm6th | Unicode version |
Description: One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.) |
Ref | Expression |
---|---|
halfpm6th |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cn 8923 | . . . . . 6 | |
2 | ax-1cn 7837 | . . . . . 6 | |
3 | 2cn 8919 | . . . . . 6 | |
4 | 3re 8922 | . . . . . . 7 | |
5 | 3pos 8942 | . . . . . . 7 | |
6 | 4, 5 | gt0ap0ii 8517 | . . . . . 6 # |
7 | 2ap0 8941 | . . . . . 6 # | |
8 | 1, 1, 2, 3, 6, 7 | divmuldivapi 8659 | . . . . 5 |
9 | 1, 6 | dividapi 8632 | . . . . . . 7 |
10 | 9 | oveq1i 5846 | . . . . . 6 |
11 | halfcn 9062 | . . . . . . 7 | |
12 | 11 | mulid2i 7893 | . . . . . 6 |
13 | 10, 12 | eqtri 2185 | . . . . 5 |
14 | 1 | mulid1i 7892 | . . . . . 6 |
15 | 3t2e6 9004 | . . . . . 6 | |
16 | 14, 15 | oveq12i 5848 | . . . . 5 |
17 | 8, 13, 16 | 3eqtr3i 2193 | . . . 4 |
18 | 17 | oveq1i 5846 | . . 3 |
19 | 6cn 8930 | . . . . 5 | |
20 | 6re 8929 | . . . . . 6 | |
21 | 6pos 8949 | . . . . . 6 | |
22 | 20, 21 | gt0ap0ii 8517 | . . . . 5 # |
23 | 19, 22 | pm3.2i 270 | . . . 4 # |
24 | divsubdirap 8595 | . . . 4 # | |
25 | 1, 2, 23, 24 | mp3an 1326 | . . 3 |
26 | 3m1e2 8968 | . . . . 5 | |
27 | 26 | oveq1i 5846 | . . . 4 |
28 | 3 | mulid2i 7893 | . . . . 5 |
29 | 28, 15 | oveq12i 5848 | . . . 4 |
30 | 3, 7 | dividapi 8632 | . . . . . 6 |
31 | 30 | oveq2i 5847 | . . . . 5 |
32 | 2, 1, 3, 3, 6, 7 | divmuldivapi 8659 | . . . . 5 |
33 | 1, 6 | recclapi 8629 | . . . . . 6 |
34 | 33 | mulid1i 7892 | . . . . 5 |
35 | 31, 32, 34 | 3eqtr3i 2193 | . . . 4 |
36 | 27, 29, 35 | 3eqtr2i 2191 | . . 3 |
37 | 18, 25, 36 | 3eqtr2i 2191 | . 2 |
38 | 1, 2, 19, 22 | divdirapi 8656 | . . . 4 |
39 | df-4 8909 | . . . . 5 | |
40 | 39 | oveq1i 5846 | . . . 4 |
41 | 17 | oveq1i 5846 | . . . 4 |
42 | 38, 40, 41 | 3eqtr4ri 2196 | . . 3 |
43 | 2t2e4 9002 | . . . 4 | |
44 | 43, 15 | oveq12i 5848 | . . 3 |
45 | 30 | oveq2i 5847 | . . . 4 |
46 | 3, 1, 3, 3, 6, 7 | divmuldivapi 8659 | . . . 4 |
47 | 3, 1, 6 | divclapi 8641 | . . . . 5 |
48 | 47 | mulid1i 7892 | . . . 4 |
49 | 45, 46, 48 | 3eqtr3i 2193 | . . 3 |
50 | 42, 44, 49 | 3eqtr2i 2191 | . 2 |
51 | 37, 50 | pm3.2i 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 cc 7742 cc0 7744 c1 7745 caddc 7747 cmul 7749 cmin 8060 # cap 8470 cdiv 8559 c2 8899 c3 8900 c4 8901 c6 8903 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 |
This theorem is referenced by: cos01bnd 11685 |
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