Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > flodddiv4t2lthalf | Unicode version |
Description: The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.) |
Ref | Expression |
---|---|
flodddiv4t2lthalf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flodddiv4lt 10336 | . . 3 | |
2 | 4nn 8195 | . . . . . . . 8 | |
3 | znq 8709 | . . . . . . . 8 | |
4 | 2, 3 | mpan2 415 | . . . . . . 7 |
5 | 4 | flqcld 9279 | . . . . . 6 |
6 | 5 | zred 8469 | . . . . 5 |
7 | 6 | adantr 270 | . . . 4 |
8 | qre 8710 | . . . . . 6 | |
9 | 4, 8 | syl 14 | . . . . 5 |
10 | 9 | adantr 270 | . . . 4 |
11 | 2re 8109 | . . . . . 6 | |
12 | 2pos 8130 | . . . . . 6 | |
13 | 11, 12 | pm3.2i 266 | . . . . 5 |
14 | 13 | a1i 9 | . . . 4 |
15 | ltmul1 7692 | . . . 4 | |
16 | 7, 10, 14, 15 | syl3anc 1169 | . . 3 |
17 | 1, 16 | mpbid 145 | . 2 |
18 | zcn 8356 | . . . . . 6 | |
19 | 18 | halfcld 8275 | . . . . 5 |
20 | 2cnd 8112 | . . . . 5 | |
21 | 2ap0 8132 | . . . . . 6 # | |
22 | 21 | a1i 9 | . . . . 5 # |
23 | 19, 20, 22 | divcanap1d 7878 | . . . 4 |
24 | 18, 20, 20, 22, 22 | divdivap1d 7908 | . . . . . 6 |
25 | 2t2e4 8186 | . . . . . . . 8 | |
26 | 25 | a1i 9 | . . . . . . 7 |
27 | 26 | oveq2d 5548 | . . . . . 6 |
28 | 24, 27 | eqtrd 2113 | . . . . 5 |
29 | 28 | oveq1d 5547 | . . . 4 |
30 | 23, 29 | eqtr3d 2115 | . . 3 |
31 | 30 | adantr 270 | . 2 |
32 | 17, 31 | breqtrrd 3811 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 class class class wbr 3785 cfv 4922 (class class class)co 5532 cr 6980 cc0 6981 cmul 6986 clt 7153 # cap 7681 cdiv 7760 cn 8039 c2 8089 c4 8091 cz 8351 cq 8704 cfl 9272 cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-q 8705 df-rp 8735 df-fl 9274 df-dvds 10196 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |