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Mirrors > Home > ILE Home > Th. List > nndivdvds | Unicode version |
Description: Strong form of dvdsval2 11933 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
nndivdvds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9336 |
. . . . 5
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2 | 1 | adantl 277 |
. . . 4
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3 | nnne0 9010 |
. . . . 5
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4 | 3 | adantl 277 |
. . . 4
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5 | nnz 9336 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | adantr 276 |
. . . 4
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7 | dvdsval2 11933 |
. . . 4
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8 | 2, 4, 6, 7 | syl3anc 1249 |
. . 3
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9 | 8 | anbi1d 465 |
. 2
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10 | nnre 8989 |
. . . . 5
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11 | 10 | adantr 276 |
. . . 4
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12 | nnre 8989 |
. . . . 5
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13 | 12 | adantl 277 |
. . . 4
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14 | nngt0 9007 |
. . . . 5
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15 | 14 | adantr 276 |
. . . 4
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16 | nngt0 9007 |
. . . . 5
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17 | 16 | adantl 277 |
. . . 4
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18 | 11, 13, 15, 17 | divgt0d 8954 |
. . 3
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19 | 18 | biantrud 304 |
. 2
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20 | elnnz 9327 |
. . 3
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21 | 20 | a1i 9 |
. 2
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22 | 9, 19, 21 | 3bitr4d 220 |
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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
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 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-dvds 11931 |
This theorem is referenced by: nndivides 11940 dvdsdivcl 11992 divgcdnn 12112 lcmgcdlem 12215 isprm6 12285 oddpwdclemodd 12310 oddpwdclemdc 12311 divnumden 12334 hashgcdlem 12376 hashgcdeq 12377 oddprmdvds 12492 infpnlem2 12498 infpn2 12613 znrrg 14148 |
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