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 3980 | . . . . . . . 8 | |
3 | breq1 3980 | . . . . . . . . 9 | |
4 | breq1 3980 | . . . . . . . . 9 | |
5 | 3, 4 | anbi12d 465 | . . . . . . . 8 |
6 | 2, 5 | bibi12d 234 | . . . . . . 7 |
7 | equcom 1693 | . . . . . . 7 | |
8 | bicom 139 | . . . . . . 7 | |
9 | 6, 7, 8 | 3imtr3i 199 | . . . . . 6 |
10 | bezoutlemgcd.4 | . . . . . . . 8 | |
11 | 6 | cbvralv 2690 | . . . . . . . 8 |
12 | 10, 11 | sylib 121 | . . . . . . 7 |
13 | 12 | ad2antrr 480 | . . . . . 6 |
14 | simplr 520 | . . . . . 6 | |
15 | 9, 13, 14 | rspcdva 2831 | . . . . 5 |
16 | 1, 15 | mpbird 166 | . . . 4 |
17 | bezoutlemgcd.3 | . . . . . . 7 | |
18 | 17 | ad2antrr 480 | . . . . . 6 |
19 | bezoutlemgcd.5 | . . . . . . . . 9 | |
20 | 19 | ad2antrr 480 | . . . . . . . 8 |
21 | breq1 3980 | . . . . . . . . . . . 12 | |
22 | breq1 3980 | . . . . . . . . . . . . 13 | |
23 | breq1 3980 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | anbi12d 465 | . . . . . . . . . . . 12 |
25 | 21, 24 | bibi12d 234 | . . . . . . . . . . 11 |
26 | 0zd 9195 | . . . . . . . . . . 11 | |
27 | 25, 10, 26 | rspcdva 2831 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 480 | . . . . . . . . 9 |
29 | 18 | nn0zd 9303 | . . . . . . . . . 10 |
30 | 0dvds 11738 | . . . . . . . . . 10 | |
31 | 29, 30 | syl 14 | . . . . . . . . 9 |
32 | bezoutlemgcd.1 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antrr 480 | . . . . . . . . . . 11 |
34 | 0dvds 11738 | . . . . . . . . . . 11 | |
35 | 33, 34 | syl 14 | . . . . . . . . . 10 |
36 | bezoutlemgcd.2 | . . . . . . . . . . . 12 | |
37 | 36 | ad2antrr 480 | . . . . . . . . . . 11 |
38 | 0dvds 11738 | . . . . . . . . . . 11 | |
39 | 37, 38 | syl 14 | . . . . . . . . . 10 |
40 | 35, 39 | anbi12d 465 | . . . . . . . . 9 |
41 | 28, 31, 40 | 3bitr3d 217 | . . . . . . . 8 |
42 | 20, 41 | mtbird 663 | . . . . . . 7 |
43 | 42 | neqned 2341 | . . . . . 6 |
44 | elnnne0 9120 | . . . . . 6 | |
45 | 18, 43, 44 | sylanbrc 414 | . . . . 5 |
46 | dvdsle 11768 | . . . . 5 | |
47 | 14, 45, 46 | syl2anc 409 | . . . 4 |
48 | 16, 47 | mpd 13 | . . 3 |
49 | 48 | ex 114 | . 2 |
50 | 49 | ralrimiva 2537 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wne 2334 wral 2442 class class class wbr 3977 cc0 7745 cle 7926 cn 8849 cn0 9106 cz 9183 cdvds 11714 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-po 4269 df-iso 4270 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-n0 9107 df-z 9184 df-q 9550 df-dvds 11715 |
This theorem is referenced by: bezoutlemsup 11928 |
Copyright terms: Public domain | W3C validator |