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Mirrors > Home > ILE Home > Th. List > 2tnp1ge0ge0 | Unicode version |
Description: Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
2tnp1ge0ge0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9312 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
2 | 1 | a1i 9 |
. . . . . 6
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3 | id 19 |
. . . . . 6
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4 | 2, 3 | zmulcld 9412 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | peano2zd 9409 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | zred 9406 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 2re 9020 |
. . . 4
![]() ![]() ![]() ![]() | |
8 | 7 | a1i 9 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 2pos 9041 |
. . . 4
![]() ![]() ![]() ![]() | |
10 | 9 | a1i 9 |
. . 3
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11 | ge0div 8859 |
. . 3
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12 | 6, 8, 10, 11 | syl3anc 1249 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 4 | zcnd 9407 |
. . . . 5
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14 | 1cnd 8004 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 2cn 9021 |
. . . . . . 7
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16 | 2ap0 9043 |
. . . . . . 7
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17 | 15, 16 | pm3.2i 272 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | a1i 9 |
. . . . 5
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19 | divdirap 8685 |
. . . . 5
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20 | 13, 14, 18, 19 | syl3anc 1249 |
. . . 4
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21 | zcn 9289 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 2cnd 9023 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 16 | a1i 9 |
. . . . . 6
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24 | 21, 22, 23 | divcanap3d 8783 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | oveq1d 5912 |
. . . 4
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26 | 20, 25 | eqtrd 2222 |
. . 3
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27 | 26 | breq2d 4030 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | zre 9288 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | halfre 9163 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 28, 30 | readdcld 8018 |
. . 3
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32 | halfge0 9166 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 28, 30 | addge01d 8521 |
. . . 4
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34 | 32, 33 | mpbii 148 |
. . 3
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35 | 1red 8003 |
. . . 4
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36 | halflt1 9167 |
. . . . 5
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37 | 36 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 30, 35, 28, 37 | ltadd2dd 8410 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | btwnzge0 10333 |
. . 3
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40 | 31, 3, 34, 38, 39 | syl22anc 1250 |
. 2
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41 | 12, 27, 40 | 3bitrd 214 |
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-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 |
This theorem is referenced by: oddnn02np1 11920 |
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