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Mirrors > Home > ILE Home > Th. List > gcd0id | Unicode version |
Description: The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcd0id |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcd0val 12002 |
. . . 4
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2 | oveq2 5908 |
. . . 4
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3 | fveq2 5537 |
. . . . 5
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4 | abs0 11108 |
. . . . 5
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5 | 3, 4 | eqtrdi 2238 |
. . . 4
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6 | 1, 2, 5 | 3eqtr4a 2248 |
. . 3
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7 | 6 | adantl 277 |
. 2
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8 | df-ne 2361 |
. . 3
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9 | 0z 9299 |
. . . . . . . 8
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10 | gcddvds 12005 |
. . . . . . . 8
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11 | 9, 10 | mpan 424 |
. . . . . . 7
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12 | 11 | simprd 114 |
. . . . . 6
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13 | 12 | adantr 276 |
. . . . 5
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14 | gcdcl 12008 |
. . . . . . . . 9
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15 | 9, 14 | mpan 424 |
. . . . . . . 8
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16 | 15 | nn0zd 9408 |
. . . . . . 7
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17 | dvdsleabs 11892 |
. . . . . . 7
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18 | 16, 17 | syl3an1 1282 |
. . . . . 6
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19 | 18 | 3anidm12 1306 |
. . . . 5
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20 | 13, 19 | mpd 13 |
. . . 4
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21 | zabscl 11136 |
. . . . . . . 8
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22 | dvds0 11854 |
. . . . . . . 8
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23 | 21, 22 | syl 14 |
. . . . . . 7
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24 | iddvds 11852 |
. . . . . . . 8
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25 | absdvdsb 11857 |
. . . . . . . . 9
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26 | 25 | anidms 397 |
. . . . . . . 8
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27 | 24, 26 | mpbid 147 |
. . . . . . 7
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28 | 23, 27 | jca 306 |
. . . . . 6
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29 | 28 | adantr 276 |
. . . . 5
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30 | eqid 2189 |
. . . . . . . . 9
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31 | 30 | biantrur 303 |
. . . . . . . 8
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32 | 31 | necon3abii 2396 |
. . . . . . 7
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33 | dvdslegcd 12006 |
. . . . . . . . . 10
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34 | 33 | ex 115 |
. . . . . . . . 9
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35 | 9, 34 | mp3an2 1336 |
. . . . . . . 8
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36 | 21, 35 | mpancom 422 |
. . . . . . 7
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37 | 32, 36 | biimtrid 152 |
. . . . . 6
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38 | 37 | imp 124 |
. . . . 5
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39 | 29, 38 | mpd 13 |
. . . 4
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40 | 16 | zred 9410 |
. . . . . 6
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41 | 21 | zred 9410 |
. . . . . 6
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42 | 40, 41 | letri3d 8108 |
. . . . 5
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43 | 42 | adantr 276 |
. . . 4
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44 | 20, 39, 43 | mpbir2and 946 |
. . 3
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45 | 8, 44 | sylan2br 288 |
. 2
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46 | zdceq 9363 |
. . . 4
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47 | 9, 46 | mpan2 425 |
. . 3
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48 | exmiddc 837 |
. . 3
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49 | 47, 48 | syl 14 |
. 2
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50 | 7, 45, 49 | mpjaodan 799 |
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-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 ax-caucvg 7966 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-sup 7017 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-2 9013 df-3 9014 df-4 9015 df-n0 9212 df-z 9289 df-uz 9564 df-q 9656 df-rp 9690 df-fz 10045 df-fzo 10179 df-fl 10307 df-mod 10360 df-seqfrec 10485 df-exp 10560 df-cj 10892 df-re 10893 df-im 10894 df-rsqrt 11048 df-abs 11049 df-dvds 11836 df-gcd 11985 |
This theorem is referenced by: gcdid0 12022 nn0gcdsq 12243 dfphi2 12263 |
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