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Mirrors > Home > ILE Home > Th. List > bezoutlema | 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 |
---|---|
bezoutlema |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9198 | . . 3 | |
2 | 0z 9183 | . . 3 | |
3 | bezoutlema.b | . . . . . . 7 | |
4 | 3 | nn0cnd 9150 | . . . . . 6 |
5 | 4 | mul01d 8272 | . . . . 5 |
6 | 5 | oveq2d 5842 | . . . 4 |
7 | bezoutlema.a | . . . . . . 7 | |
8 | 7 | nn0cnd 9150 | . . . . . 6 |
9 | 1cnd 7896 | . . . . . 6 | |
10 | 8, 9 | mulcld 7900 | . . . . 5 |
11 | 10 | addid1d 8028 | . . . 4 |
12 | 8 | mulid1d 7897 | . . . 4 |
13 | 6, 11, 12 | 3eqtrrd 2195 | . . 3 |
14 | oveq2 5834 | . . . . . 6 | |
15 | 14 | oveq1d 5841 | . . . . 5 |
16 | 15 | eqeq2d 2169 | . . . 4 |
17 | oveq2 5834 | . . . . . 6 | |
18 | 17 | oveq2d 5842 | . . . . 5 |
19 | 18 | eqeq2d 2169 | . . . 4 |
20 | 16, 19 | rspc2ev 2831 | . . 3 |
21 | 1, 2, 13, 20 | mp3an12i 1323 | . 2 |
22 | bezoutlema.is-bezout | . . . . 5 | |
23 | eqeq1 2164 | . . . . . 6 | |
24 | 23 | 2rexbidv 2482 | . . . . 5 |
25 | 22, 24 | syl5bb 191 | . . . 4 |
26 | 25 | sbcieg 2969 | . . 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 1335 wcel 2128 wrex 2436 wsbc 2937 (class class class)co 5826 cc0 7734 c1 7735 caddc 7737 cmul 7739 cn0 9095 cz 9172 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-n0 9096 df-z 9173 |
This theorem is referenced by: bezoutlemex 11900 |
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