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Mirrors > Home > ILE Home > Th. List > bezoutlemle | Unicode version |
Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both and . (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) |
Ref | Expression |
---|---|
bezoutlemgcd.1 | |
bezoutlemgcd.2 | |
bezoutlemgcd.3 | |
bezoutlemgcd.4 | |
bezoutlemgcd.5 |
Ref | Expression |
---|---|
bezoutlemle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | breq1 3902 | . . . . . . . 8 | |
3 | breq1 3902 | . . . . . . . . 9 | |
4 | breq1 3902 | . . . . . . . . 9 | |
5 | 3, 4 | anbi12d 464 | . . . . . . . 8 |
6 | 2, 5 | bibi12d 234 | . . . . . . 7 |
7 | equcom 1667 | . . . . . . 7 | |
8 | bicom 139 | . . . . . . 7 | |
9 | 6, 7, 8 | 3imtr3i 199 | . . . . . 6 |
10 | bezoutlemgcd.4 | . . . . . . . 8 | |
11 | 6 | cbvralv 2631 | . . . . . . . 8 |
12 | 10, 11 | sylib 121 | . . . . . . 7 |
13 | 12 | ad2antrr 479 | . . . . . 6 |
14 | simplr 504 | . . . . . 6 | |
15 | 9, 13, 14 | rspcdva 2768 | . . . . 5 |
16 | 1, 15 | mpbird 166 | . . . 4 |
17 | bezoutlemgcd.3 | . . . . . . 7 | |
18 | 17 | ad2antrr 479 | . . . . . 6 |
19 | bezoutlemgcd.5 | . . . . . . . . 9 | |
20 | 19 | ad2antrr 479 | . . . . . . . 8 |
21 | breq1 3902 | . . . . . . . . . . . 12 | |
22 | breq1 3902 | . . . . . . . . . . . . 13 | |
23 | breq1 3902 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | anbi12d 464 | . . . . . . . . . . . 12 |
25 | 21, 24 | bibi12d 234 | . . . . . . . . . . 11 |
26 | 0zd 9024 | . . . . . . . . . . 11 | |
27 | 25, 10, 26 | rspcdva 2768 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 479 | . . . . . . . . 9 |
29 | 18 | nn0zd 9129 | . . . . . . . . . 10 |
30 | 0dvds 11425 | . . . . . . . . . 10 | |
31 | 29, 30 | syl 14 | . . . . . . . . 9 |
32 | bezoutlemgcd.1 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antrr 479 | . . . . . . . . . . 11 |
34 | 0dvds 11425 | . . . . . . . . . . 11 | |
35 | 33, 34 | syl 14 | . . . . . . . . . 10 |
36 | bezoutlemgcd.2 | . . . . . . . . . . . 12 | |
37 | 36 | ad2antrr 479 | . . . . . . . . . . 11 |
38 | 0dvds 11425 | . . . . . . . . . . 11 | |
39 | 37, 38 | syl 14 | . . . . . . . . . 10 |
40 | 35, 39 | anbi12d 464 | . . . . . . . . 9 |
41 | 28, 31, 40 | 3bitr3d 217 | . . . . . . . 8 |
42 | 20, 41 | mtbird 647 | . . . . . . 7 |
43 | 42 | neqned 2292 | . . . . . 6 |
44 | elnnne0 8949 | . . . . . 6 | |
45 | 18, 43, 44 | sylanbrc 413 | . . . . 5 |
46 | dvdsle 11454 | . . . . 5 | |
47 | 14, 45, 46 | syl2anc 408 | . . . 4 |
48 | 16, 47 | mpd 13 | . . 3 |
49 | 48 | ex 114 | . 2 |
50 | 49 | ralrimiva 2482 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wne 2285 wral 2393 class class class wbr 3899 cc0 7588 cle 7769 cn 8684 cn0 8935 cz 9012 cdvds 11405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-n0 8936 df-z 9013 df-q 9368 df-dvds 11406 |
This theorem is referenced by: bezoutlemsup 11609 |
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