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Mirrors > Home > ILE Home > Th. List > oddpwdc | Unicode version |
Description: The function ![]() |
Ref | Expression |
---|---|
oddpwdc.j |
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oddpwdc.f |
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Ref | Expression |
---|---|
oddpwdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddpwdc.f |
. . 3
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2 | 2cnd 8697 |
. . . . . 6
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3 | simpr 109 |
. . . . . 6
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4 | 2, 3 | expcld 10311 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | breq2 3897 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | notbid 639 |
. . . . . . . . 9
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7 | oddpwdc.j |
. . . . . . . . 9
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8 | 6, 7 | elrab2 2810 |
. . . . . . . 8
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9 | 8 | simplbi 270 |
. . . . . . 7
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10 | 9 | adantr 272 |
. . . . . 6
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11 | 10 | nncnd 8638 |
. . . . 5
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12 | 4, 11 | mulcld 7704 |
. . . 4
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13 | 12 | adantl 273 |
. . 3
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14 | nnnn0 8882 |
. . . . . 6
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15 | 2nn 8779 |
. . . . . . 7
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16 | pw2dvdseu 11685 |
. . . . . . . 8
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17 | riotacl 5696 |
. . . . . . . 8
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18 | 16, 17 | syl 14 |
. . . . . . 7
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19 | nnexpcl 10193 |
. . . . . . 7
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20 | 15, 18, 19 | sylancr 408 |
. . . . . 6
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21 | nn0nndivcl 8937 |
. . . . . 6
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22 | 14, 20, 21 | syl2anc 406 |
. . . . 5
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23 | 22, 18 | jca 302 |
. . . 4
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24 | 23 | adantl 273 |
. . 3
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25 | 8 | anbi1i 451 |
. . . . . 6
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26 | 25 | anbi1i 451 |
. . . . 5
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27 | oddpwdclemdc 11690 |
. . . . 5
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28 | 26, 27 | bitri 183 |
. . . 4
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29 | 28 | a1i 9 |
. . 3
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30 | 1, 13, 24, 29 | f1od2 6084 |
. 2
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31 | 30 | mptru 1321 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 ax-arch 7658 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-frec 6240 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-2 8683 df-n0 8876 df-z 8953 df-uz 9223 df-q 9308 df-rp 9338 df-fz 9678 df-fl 9930 df-mod 9983 df-seqfrec 10106 df-exp 10180 df-dvds 11336 |
This theorem is referenced by: sqpweven 11692 2sqpwodd 11693 xpnnen 11746 |
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