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Mirrors > Home > ILE Home > Th. List > ncoprmgcdne1b | Unicode version |
Description: Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.) |
Ref | Expression |
---|---|
ncoprmgcdne1b |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8974 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 2re 8985 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
3 | 2 | a1i 9 |
. . . . . . . 8
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4 | eluzelz 9533 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | ad2antlr 489 |
. . . . . . . . 9
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6 | 5 | zred 9371 |
. . . . . . . 8
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7 | simplll 533 |
. . . . . . . . . 10
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8 | simpllr 534 |
. . . . . . . . . 10
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9 | gcdnncl 11960 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 7, 8, 9 | syl2anc 411 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | nnred 8928 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | eluzle 9536 |
. . . . . . . . 9
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13 | 12 | ad2antlr 489 |
. . . . . . . 8
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14 | simpr 110 |
. . . . . . . . . 10
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15 | 7 | nnzd 9370 |
. . . . . . . . . . 11
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16 | 8 | nnzd 9370 |
. . . . . . . . . . 11
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17 | dvdsgcd 12005 |
. . . . . . . . . . 11
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18 | 5, 15, 16, 17 | syl3anc 1238 |
. . . . . . . . . 10
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19 | 14, 18 | mpd 13 |
. . . . . . . . 9
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20 | dvdsle 11842 |
. . . . . . . . . 10
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21 | 5, 10, 20 | syl2anc 411 |
. . . . . . . . 9
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22 | 19, 21 | mpd 13 |
. . . . . . . 8
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23 | 3, 6, 11, 13, 22 | letrd 8077 |
. . . . . . 7
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24 | 1, 23 | eqbrtrrid 4038 |
. . . . . 6
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25 | 1nn 8926 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
26 | 25 | a1i 9 |
. . . . . . 7
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27 | nnltp1le 9309 |
. . . . . . 7
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28 | 26, 10, 27 | syl2anc 411 |
. . . . . 6
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29 | 24, 28 | mpbird 167 |
. . . . 5
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30 | nngt1ne1 8950 |
. . . . . 6
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31 | 10, 30 | syl 14 |
. . . . 5
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32 | 29, 31 | mpbid 147 |
. . . 4
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33 | 32 | ex 115 |
. . 3
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34 | 33 | rexlimdva 2594 |
. 2
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35 | 9 | adantr 276 |
. . . . 5
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36 | simpr 110 |
. . . . 5
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37 | eluz2b3 9600 |
. . . . 5
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38 | 35, 36, 37 | sylanbrc 417 |
. . . 4
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39 | simpll 527 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 39 | nnzd 9370 |
. . . . 5
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41 | simplr 528 |
. . . . . 6
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42 | 41 | nnzd 9370 |
. . . . 5
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43 | gcddvds 11956 |
. . . . 5
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44 | 40, 42, 43 | syl2anc 411 |
. . . 4
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45 | breq1 4005 |
. . . . . 6
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46 | breq1 4005 |
. . . . . 6
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47 | 45, 46 | anbi12d 473 |
. . . . 5
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48 | 47 | rspcev 2841 |
. . . 4
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49 | 38, 44, 48 | syl2anc 411 |
. . 3
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50 | 49 | ex 115 |
. 2
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51 | 34, 50 | impbid 129 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 ax-caucvg 7928 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-sup 6980 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-n0 9173 df-z 9250 df-uz 9525 df-q 9616 df-rp 9650 df-fz 10005 df-fzo 10138 df-fl 10265 df-mod 10318 df-seqfrec 10441 df-exp 10515 df-cj 10844 df-re 10845 df-im 10846 df-rsqrt 11000 df-abs 11001 df-dvds 11788 df-gcd 11936 |
This theorem is referenced by: ncoprmgcdgt1b 12082 |
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