Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3932 | . . . . . . . 8 | |
3 | breq1 3932 | . . . . . . . . 9 | |
4 | breq1 3932 | . . . . . . . . 9 | |
5 | 3, 4 | anbi12d 464 | . . . . . . . 8 |
6 | 2, 5 | bibi12d 234 | . . . . . . 7 |
7 | equcom 1682 | . . . . . . 7 | |
8 | bicom 139 | . . . . . . 7 | |
9 | 6, 7, 8 | 3imtr3i 199 | . . . . . 6 |
10 | bezoutlemgcd.4 | . . . . . . . 8 | |
11 | 6 | cbvralv 2654 | . . . . . . . 8 |
12 | 10, 11 | sylib 121 | . . . . . . 7 |
13 | 12 | ad2antrr 479 | . . . . . 6 |
14 | simplr 519 | . . . . . 6 | |
15 | 9, 13, 14 | rspcdva 2794 | . . . . 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 3932 | . . . . . . . . . . . 12 | |
22 | breq1 3932 | . . . . . . . . . . . . 13 | |
23 | breq1 3932 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | anbi12d 464 | . . . . . . . . . . . 12 |
25 | 21, 24 | bibi12d 234 | . . . . . . . . . . 11 |
26 | 0zd 9066 | . . . . . . . . . . 11 | |
27 | 25, 10, 26 | rspcdva 2794 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 479 | . . . . . . . . 9 |
29 | 18 | nn0zd 9171 | . . . . . . . . . 10 |
30 | 0dvds 11513 | . . . . . . . . . 10 | |
31 | 29, 30 | syl 14 | . . . . . . . . 9 |
32 | bezoutlemgcd.1 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antrr 479 | . . . . . . . . . . 11 |
34 | 0dvds 11513 | . . . . . . . . . . 11 | |
35 | 33, 34 | syl 14 | . . . . . . . . . 10 |
36 | bezoutlemgcd.2 | . . . . . . . . . . . 12 | |
37 | 36 | ad2antrr 479 | . . . . . . . . . . 11 |
38 | 0dvds 11513 | . . . . . . . . . . 11 | |
39 | 37, 38 | syl 14 | . . . . . . . . . 10 |
40 | 35, 39 | anbi12d 464 | . . . . . . . . 9 |
41 | 28, 31, 40 | 3bitr3d 217 | . . . . . . . 8 |
42 | 20, 41 | mtbird 662 | . . . . . . 7 |
43 | 42 | neqned 2315 | . . . . . 6 |
44 | elnnne0 8991 | . . . . . 6 | |
45 | 18, 43, 44 | sylanbrc 413 | . . . . 5 |
46 | dvdsle 11542 | . . . . 5 | |
47 | 14, 45, 46 | syl2anc 408 | . . . 4 |
48 | 16, 47 | mpd 13 | . . 3 |
49 | 48 | ex 114 | . 2 |
50 | 49 | ralrimiva 2505 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2308 wral 2416 class class class wbr 3929 cc0 7620 cle 7801 cn 8720 cn0 8977 cz 9054 cdvds 11493 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-q 9412 df-dvds 11494 |
This theorem is referenced by: bezoutlemsup 11697 |
Copyright terms: Public domain | W3C validator |