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Mirrors > Home > ILE Home > Th. List > oddpwdclemxy | Unicode version |
Description: Lemma for oddpwdc 12141. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Ref | Expression |
---|---|
oddpwdclemxy |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 9053 | . . . . . 6 | |
2 | 1 | a1i 9 | . . . . 5 |
3 | simplll 533 | . . . . . . . . 9 | |
4 | 3 | nnzd 9347 | . . . . . . . 8 |
5 | simplr 528 | . . . . . . . . . 10 | |
6 | 2, 5 | nnexpcld 10645 | . . . . . . . . 9 |
7 | 6 | nnzd 9347 | . . . . . . . 8 |
8 | simpr 110 | . . . . . . . . . 10 | |
9 | 6, 3 | nnmulcld 8941 | . . . . . . . . . 10 |
10 | 8, 9 | eqeltrd 2252 | . . . . . . . . 9 |
11 | 10 | nnzd 9347 | . . . . . . . 8 |
12 | 6 | nncnd 8906 | . . . . . . . . . 10 |
13 | 3 | nncnd 8906 | . . . . . . . . . 10 |
14 | 12, 13 | mulcomd 7953 | . . . . . . . . 9 |
15 | 8, 14 | eqtr2d 2209 | . . . . . . . 8 |
16 | dvds0lem 11776 | . . . . . . . 8 | |
17 | 4, 7, 11, 15, 16 | syl31anc 1241 | . . . . . . 7 |
18 | simpllr 534 | . . . . . . . . 9 | |
19 | 8 | breq2d 4010 | . . . . . . . . . 10 |
20 | 2 | nnzd 9347 | . . . . . . . . . . 11 |
21 | 6 | nnne0d 8937 | . . . . . . . . . . 11 |
22 | dvdscmulr 11795 | . . . . . . . . . . 11 | |
23 | 20, 4, 7, 21, 22 | syl112anc 1242 | . . . . . . . . . 10 |
24 | 19, 23 | bitrd 188 | . . . . . . . . 9 |
25 | 18, 24 | mtbird 673 | . . . . . . . 8 |
26 | 2 | nncnd 8906 | . . . . . . . . . 10 |
27 | 26, 5 | expp1d 10624 | . . . . . . . . 9 |
28 | 27 | breq1d 4008 | . . . . . . . 8 |
29 | 25, 28 | mtbird 673 | . . . . . . 7 |
30 | pw2dvdseu 12135 | . . . . . . . . 9 | |
31 | 10, 30 | syl 14 | . . . . . . . 8 |
32 | oveq2 5873 | . . . . . . . . . . 11 | |
33 | 32 | breq1d 4008 | . . . . . . . . . 10 |
34 | oveq1 5872 | . . . . . . . . . . . . 13 | |
35 | 34 | oveq2d 5881 | . . . . . . . . . . . 12 |
36 | 35 | breq1d 4008 | . . . . . . . . . . 11 |
37 | 36 | notbid 667 | . . . . . . . . . 10 |
38 | 33, 37 | anbi12d 473 | . . . . . . . . 9 |
39 | 38 | riota2 5843 | . . . . . . . 8 |
40 | 5, 31, 39 | syl2anc 411 | . . . . . . 7 |
41 | 17, 29, 40 | mpbi2and 943 | . . . . . 6 |
42 | 41, 5 | eqeltrd 2252 | . . . . 5 |
43 | 2, 42 | nnexpcld 10645 | . . . 4 |
44 | 43 | nncnd 8906 | . . 3 |
45 | 43 | nnap0d 8938 | . . 3 # |
46 | 41 | eqcomd 2181 | . . . . . 6 |
47 | 46 | oveq2d 5881 | . . . . 5 |
48 | 47 | oveq1d 5880 | . . . 4 |
49 | 8, 48 | eqtr2d 2209 | . . 3 |
50 | 44, 13, 45, 49 | mvllmulapd 8772 | . 2 |
51 | 50, 46 | jca 306 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wne 2345 wreu 2455 class class class wbr 3998 crio 5820 (class class class)co 5865 cc0 7786 c1 7787 caddc 7789 cmul 7791 cdiv 8602 cn 8892 c2 8943 cn0 9149 cz 9226 cexp 10489 cdvds 11762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-fz 9980 df-fl 10240 df-mod 10293 df-seqfrec 10416 df-exp 10490 df-dvds 11763 |
This theorem is referenced by: oddpwdclemdc 12140 |
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