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Mirrors > Home > ILE Home > Th. List > bezoutlemb | Unicode version |
Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by . (Contributed by Jim Kingdon, 30-Dec-2021.) |
Ref | Expression |
---|---|
bezoutlema.is-bezout | |
bezoutlema.a | |
bezoutlema.b |
Ref | Expression |
---|---|
bezoutlemb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9065 | . . 3 | |
2 | 1z 9080 | . . 3 | |
3 | bezoutlema.a | . . . . . . 7 | |
4 | 3 | nn0cnd 9032 | . . . . . 6 |
5 | 4 | mul01d 8155 | . . . . 5 |
6 | 5 | oveq1d 5789 | . . . 4 |
7 | bezoutlema.b | . . . . . . 7 | |
8 | 7 | nn0cnd 9032 | . . . . . 6 |
9 | 1cnd 7782 | . . . . . 6 | |
10 | 8, 9 | mulcld 7786 | . . . . 5 |
11 | 10 | addid2d 7912 | . . . 4 |
12 | 8 | mulid1d 7783 | . . . 4 |
13 | 6, 11, 12 | 3eqtrrd 2177 | . . 3 |
14 | oveq2 5782 | . . . . . 6 | |
15 | 14 | oveq1d 5789 | . . . . 5 |
16 | 15 | eqeq2d 2151 | . . . 4 |
17 | oveq2 5782 | . . . . . 6 | |
18 | 17 | oveq2d 5790 | . . . . 5 |
19 | 18 | eqeq2d 2151 | . . . 4 |
20 | 16, 19 | rspc2ev 2804 | . . 3 |
21 | 1, 2, 13, 20 | mp3an12i 1319 | . 2 |
22 | bezoutlema.is-bezout | . . . . 5 | |
23 | eqeq1 2146 | . . . . . 6 | |
24 | 23 | 2rexbidv 2460 | . . . . 5 |
25 | 22, 24 | syl5bb 191 | . . . 4 |
26 | 25 | sbcieg 2941 | . . 3 |
27 | 7, 26 | syl 14 | . 2 |
28 | 21, 27 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wcel 1480 wrex 2417 wsbc 2909 (class class class)co 5774 cc0 7620 c1 7621 caddc 7623 cmul 7625 cn0 8977 cz 9054 |
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-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-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 |
This theorem is referenced by: bezoutlemex 11689 |
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