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Mirrors > Home > ILE Home > Th. List > elznn0 | Unicode version |
Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elznn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9285 |
. 2
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2 | elnn0 9208 |
. . . . . 6
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3 | 2 | a1i 9 |
. . . . 5
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4 | elnn0 9208 |
. . . . . 6
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5 | recn 7974 |
. . . . . . . . 9
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6 | 0cn 7979 |
. . . . . . . . 9
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7 | negcon1 8239 |
. . . . . . . . 9
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8 | 5, 6, 7 | sylancl 413 |
. . . . . . . 8
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9 | neg0 8233 |
. . . . . . . . . 10
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10 | 9 | eqeq1i 2197 |
. . . . . . . . 9
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11 | eqcom 2191 |
. . . . . . . . 9
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12 | 10, 11 | bitri 184 |
. . . . . . . 8
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13 | 8, 12 | bitrdi 196 |
. . . . . . 7
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14 | 13 | orbi2d 791 |
. . . . . 6
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15 | 4, 14 | bitrid 192 |
. . . . 5
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16 | 3, 15 | orbi12d 794 |
. . . 4
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17 | 3orass 983 |
. . . . 5
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18 | orcom 729 |
. . . . 5
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19 | orordir 775 |
. . . . 5
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20 | 17, 18, 19 | 3bitrri 207 |
. . . 4
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21 | 16, 20 | bitr2di 197 |
. . 3
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22 | 21 | pm5.32i 454 |
. 2
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23 | 1, 22 | bitri 184 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
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-ral 2473 df-rex 2474 df-reu 2475 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-br 4019 df-opab 4080 df-id 4311 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 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-neg 8161 df-n0 9207 df-z 9284 |
This theorem is referenced by: peano2z 9319 zmulcl 9336 elz2 9354 expnegzap 10585 expaddzaplem 10594 odd2np1 11910 bezoutlemzz 12035 bezoutlemaz 12036 bezoutlembz 12037 mulgz 13090 mulgdirlem 13093 mulgdir 13094 mulgass 13099 |
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