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Mirrors > Home > ILE Home > Th. List > halfleoddlt | Unicode version |
Description: An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.) |
Ref | Expression |
---|---|
halfleoddlt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odd2np1 11913 |
. . 3
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2 | 0xr 8035 |
. . . . . . . . . . . 12
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3 | 1re 7987 |
. . . . . . . . . . . . 13
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4 | 3 | rexri 8046 |
. . . . . . . . . . . 12
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5 | halfre 9163 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | rexri 8046 |
. . . . . . . . . . . 12
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7 | 2, 4, 6 | 3pm3.2i 1177 |
. . . . . . . . . . 11
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8 | halfgt0 9165 |
. . . . . . . . . . . 12
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9 | halflt1 9167 |
. . . . . . . . . . . 12
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10 | 8, 9 | pm3.2i 272 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | elioo3g 9942 |
. . . . . . . . . . 11
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12 | 7, 10, 11 | mpbir2an 944 |
. . . . . . . . . 10
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13 | zltaddlt1le 10039 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | mp3an3 1337 |
. . . . . . . . 9
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15 | zcn 9289 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | adantr 276 |
. . . . . . . . . . 11
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17 | 1cnd 8004 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 2cn 9021 |
. . . . . . . . . . . . 13
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19 | 2ap0 9043 |
. . . . . . . . . . . . 13
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20 | 18, 19 | pm3.2i 272 |
. . . . . . . . . . . 12
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21 | 20 | a1i 9 |
. . . . . . . . . . 11
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22 | muldivdirap 8695 |
. . . . . . . . . . 11
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23 | 16, 17, 21, 22 | syl3anc 1249 |
. . . . . . . . . 10
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24 | 23 | breq1d 4028 |
. . . . . . . . 9
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25 | 23 | breq1d 4028 |
. . . . . . . . 9
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26 | 14, 24, 25 | 3bitr4rd 221 |
. . . . . . . 8
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27 | oveq1 5904 |
. . . . . . . . . 10
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28 | 27 | breq1d 4028 |
. . . . . . . . 9
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29 | 27 | breq1d 4028 |
. . . . . . . . 9
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30 | 28, 29 | bibi12d 235 |
. . . . . . . 8
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31 | 26, 30 | syl5ibcom 155 |
. . . . . . 7
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32 | 31 | ex 115 |
. . . . . 6
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33 | 32 | adantl 277 |
. . . . 5
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34 | 33 | com23 78 |
. . . 4
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35 | 34 | rexlimdva 2607 |
. . 3
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36 | 1, 35 | sylbid 150 |
. 2
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37 | 36 | 3imp 1195 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-rp 9686 df-ioo 9924 df-dvds 11830 |
This theorem is referenced by: (None) |
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