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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 108 | . . . . 5 | |
2 | breq1 3788 | . . . . . . . 8 | |
3 | breq1 3788 | . . . . . . . . 9 | |
4 | breq1 3788 | . . . . . . . . 9 | |
5 | 3, 4 | anbi12d 456 | . . . . . . . 8 |
6 | 2, 5 | bibi12d 233 | . . . . . . 7 |
7 | equcom 1633 | . . . . . . 7 | |
8 | bicom 138 | . . . . . . 7 | |
9 | 6, 7, 8 | 3imtr3i 198 | . . . . . 6 |
10 | bezoutlemgcd.4 | . . . . . . . 8 | |
11 | 6 | cbvralv 2577 | . . . . . . . 8 |
12 | 10, 11 | sylib 120 | . . . . . . 7 |
13 | 12 | ad2antrr 471 | . . . . . 6 |
14 | simplr 496 | . . . . . 6 | |
15 | 9, 13, 14 | rspcdva 2707 | . . . . 5 |
16 | 1, 15 | mpbird 165 | . . . 4 |
17 | bezoutlemgcd.3 | . . . . . . 7 | |
18 | 17 | ad2antrr 471 | . . . . . 6 |
19 | bezoutlemgcd.5 | . . . . . . . . 9 | |
20 | 19 | ad2antrr 471 | . . . . . . . 8 |
21 | breq1 3788 | . . . . . . . . . . . 12 | |
22 | breq1 3788 | . . . . . . . . . . . . 13 | |
23 | breq1 3788 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | anbi12d 456 | . . . . . . . . . . . 12 |
25 | 21, 24 | bibi12d 233 | . . . . . . . . . . 11 |
26 | 0zd 8363 | . . . . . . . . . . 11 | |
27 | 25, 10, 26 | rspcdva 2707 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 471 | . . . . . . . . 9 |
29 | 18 | nn0zd 8467 | . . . . . . . . . 10 |
30 | 0dvds 10215 | . . . . . . . . . 10 | |
31 | 29, 30 | syl 14 | . . . . . . . . 9 |
32 | bezoutlemgcd.1 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antrr 471 | . . . . . . . . . . 11 |
34 | 0dvds 10215 | . . . . . . . . . . 11 | |
35 | 33, 34 | syl 14 | . . . . . . . . . 10 |
36 | bezoutlemgcd.2 | . . . . . . . . . . . 12 | |
37 | 36 | ad2antrr 471 | . . . . . . . . . . 11 |
38 | 0dvds 10215 | . . . . . . . . . . 11 | |
39 | 37, 38 | syl 14 | . . . . . . . . . 10 |
40 | 35, 39 | anbi12d 456 | . . . . . . . . 9 |
41 | 28, 31, 40 | 3bitr3d 216 | . . . . . . . 8 |
42 | 20, 41 | mtbird 630 | . . . . . . 7 |
43 | 42 | neqned 2252 | . . . . . 6 |
44 | elnnne0 8302 | . . . . . 6 | |
45 | 18, 43, 44 | sylanbrc 408 | . . . . 5 |
46 | dvdsle 10244 | . . . . 5 | |
47 | 14, 45, 46 | syl2anc 403 | . . . 4 |
48 | 16, 47 | mpd 13 | . . 3 |
49 | 48 | ex 113 | . 2 |
50 | 49 | ralrimiva 2434 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wne 2245 wral 2348 class class class wbr 3785 cc0 6981 cle 7154 cn 8039 cn0 8288 cz 8351 cdvds 10195 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-q 8705 df-dvds 10196 |
This theorem is referenced by: bezoutlemsup 10398 |
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