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Mirrors > Home > ILE Home > Th. List > bezoutr | Unicode version |
Description: Partial converse to bezout 10780. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Ref | Expression |
---|---|
bezoutr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdcl 10738 |
. . . 4
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2 | 1 | nn0zd 8762 |
. . 3
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3 | 2 | adantr 270 |
. 2
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4 | simpll 496 |
. . 3
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5 | simprl 498 |
. . 3
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6 | 4, 5 | zmulcld 8770 |
. 2
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7 | simplr 497 |
. . 3
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8 | simprr 499 |
. . 3
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9 | 7, 8 | zmulcld 8770 |
. 2
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10 | gcddvds 10735 |
. . . . 5
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11 | 10 | adantr 270 |
. . . 4
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12 | 11 | simpld 110 |
. . 3
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13 | dvdsmultr1 10614 |
. . . 4
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14 | 13 | imp 122 |
. . 3
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15 | 3, 4, 5, 12, 14 | syl31anc 1173 |
. 2
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16 | 11 | simprd 112 |
. . 3
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17 | dvdsmultr1 10614 |
. . . 4
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18 | 17 | imp 122 |
. . 3
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19 | 3, 7, 8, 16, 18 | syl31anc 1173 |
. 2
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20 | dvds2add 10610 |
. . 3
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21 | 20 | imp 122 |
. 2
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22 | 3, 6, 9, 15, 19, 21 | syl32anc 1178 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 ax-arch 7367 ax-caucvg 7368 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-sup 6586 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-3 8376 df-4 8377 df-n0 8566 df-z 8647 df-uz 8915 df-q 9000 df-rp 9030 df-fz 9320 df-fzo 9444 df-fl 9566 df-mod 9619 df-iseq 9741 df-iexp 9792 df-cj 10103 df-re 10104 df-im 10105 df-rsqrt 10258 df-abs 10259 df-dvds 10577 df-gcd 10719 |
This theorem is referenced by: bezoutr1 10802 |
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