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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 9348 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
2 | 1 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | id 19 |
. . . . . 6
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4 | 2, 3 | zmulcld 9448 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | peano2zd 9445 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | zred 9442 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 2re 9054 |
. . . 4
![]() ![]() ![]() ![]() | |
8 | 7 | a1i 9 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 2pos 9075 |
. . . 4
![]() ![]() ![]() ![]() | |
10 | 9 | a1i 9 |
. . 3
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11 | ge0div 8892 |
. . 3
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12 | 6, 8, 10, 11 | syl3anc 1249 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 4 | zcnd 9443 |
. . . . 5
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14 | 1cnd 8037 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 2cn 9055 |
. . . . . . 7
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16 | 2ap0 9077 |
. . . . . . 7
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17 | 15, 16 | pm3.2i 272 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | a1i 9 |
. . . . 5
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19 | divdirap 8718 |
. . . . 5
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20 | 13, 14, 18, 19 | syl3anc 1249 |
. . . 4
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21 | zcn 9325 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 2cnd 9057 |
. . . . . 6
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23 | 16 | a1i 9 |
. . . . . 6
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24 | 21, 22, 23 | divcanap3d 8816 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | oveq1d 5934 |
. . . 4
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26 | 20, 25 | eqtrd 2226 |
. . 3
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27 | 26 | breq2d 4042 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | zre 9324 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | halfre 9198 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 28, 30 | readdcld 8051 |
. . 3
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32 | halfge0 9201 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 28, 30 | addge01d 8554 |
. . . 4
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34 | 32, 33 | mpbii 148 |
. . 3
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35 | 1red 8036 |
. . . 4
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36 | halflt1 9202 |
. . . . 5
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37 | 36 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 30, 35, 28, 37 | ltadd2dd 8443 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | btwnzge0 10372 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
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 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-n0 9244 df-z 9321 |
This theorem is referenced by: oddnn02np1 12024 |
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