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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 11212 |
. . 3
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2 | 0xr 7595 |
. . . . . . . . . . . 12
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3 | 1re 7548 |
. . . . . . . . . . . . 13
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4 | 3 | rexri 7606 |
. . . . . . . . . . . 12
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5 | halfre 8690 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | rexri 7606 |
. . . . . . . . . . . 12
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7 | 2, 4, 6 | 3pm3.2i 1122 |
. . . . . . . . . . 11
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8 | halfgt0 8692 |
. . . . . . . . . . . 12
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9 | halflt1 8694 |
. . . . . . . . . . . 12
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10 | 8, 9 | pm3.2i 267 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | elioo3g 9389 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 7, 10, 11 | mpbir2an 889 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | zltaddlt1le 9484 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | mp3an3 1263 |
. . . . . . . . 9
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15 | zcn 8816 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | adantr 271 |
. . . . . . . . . . 11
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17 | 1cnd 7565 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 2cn 8554 |
. . . . . . . . . . . . 13
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19 | 2ap0 8576 |
. . . . . . . . . . . . 13
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20 | 18, 19 | pm3.2i 267 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | a1i 9 |
. . . . . . . . . . 11
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22 | muldivdirap 8235 |
. . . . . . . . . . 11
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23 | 16, 17, 21, 22 | syl3anc 1175 |
. . . . . . . . . 10
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24 | 23 | breq1d 3861 |
. . . . . . . . 9
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25 | 23 | breq1d 3861 |
. . . . . . . . 9
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26 | 14, 24, 25 | 3bitr4rd 220 |
. . . . . . . 8
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27 | oveq1 5673 |
. . . . . . . . . 10
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28 | 27 | breq1d 3861 |
. . . . . . . . 9
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29 | 27 | breq1d 3861 |
. . . . . . . . 9
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30 | 28, 29 | bibi12d 234 |
. . . . . . . 8
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31 | 26, 30 | syl5ibcom 154 |
. . . . . . 7
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32 | 31 | ex 114 |
. . . . . 6
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33 | 32 | adantl 272 |
. . . . 5
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34 | 33 | com23 78 |
. . . 4
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35 | 34 | rexlimdva 2490 |
. . 3
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36 | 1, 35 | sylbid 149 |
. 2
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37 | 36 | 3imp 1138 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-xor 1313 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 df-2 8542 df-n0 8735 df-z 8812 df-rp 9196 df-ioo 9371 df-dvds 11136 |
This theorem is referenced by: (None) |
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