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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 9089 | . . 3 | |
2 | 1z 9104 | . . 3 | |
3 | bezoutlema.a | . . . . . . 7 | |
4 | 3 | nn0cnd 9056 | . . . . . 6 |
5 | 4 | mul01d 8179 | . . . . 5 |
6 | 5 | oveq1d 5797 | . . . 4 |
7 | bezoutlema.b | . . . . . . 7 | |
8 | 7 | nn0cnd 9056 | . . . . . 6 |
9 | 1cnd 7806 | . . . . . 6 | |
10 | 8, 9 | mulcld 7810 | . . . . 5 |
11 | 10 | addid2d 7936 | . . . 4 |
12 | 8 | mulid1d 7807 | . . . 4 |
13 | 6, 11, 12 | 3eqtrrd 2178 | . . 3 |
14 | oveq2 5790 | . . . . . 6 | |
15 | 14 | oveq1d 5797 | . . . . 5 |
16 | 15 | eqeq2d 2152 | . . . 4 |
17 | oveq2 5790 | . . . . . 6 | |
18 | 17 | oveq2d 5798 | . . . . 5 |
19 | 18 | eqeq2d 2152 | . . . 4 |
20 | 16, 19 | rspc2ev 2808 | . . 3 |
21 | 1, 2, 13, 20 | mp3an12i 1320 | . 2 |
22 | bezoutlema.is-bezout | . . . . 5 | |
23 | eqeq1 2147 | . . . . . 6 | |
24 | 23 | 2rexbidv 2463 | . . . . 5 |
25 | 22, 24 | syl5bb 191 | . . . 4 |
26 | 25 | sbcieg 2945 | . . 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 1332 wcel 1481 wrex 2418 wsbc 2913 (class class class)co 5782 cc0 7644 c1 7645 caddc 7647 cmul 7649 cn0 9001 cz 9078 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: bezoutlemex 11725 |
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