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Mirrors > Home > ILE Home > Th. List > sqpweven | Unicode version |
Description: The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Ref | Expression |
---|---|
oddpwdc.j | |
oddpwdc.f |
Ref | Expression |
---|---|
sqpweven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddpwdc.j | . . . . . . . 8 | |
2 | oddpwdc.f | . . . . . . . 8 | |
3 | 1, 2 | oddpwdc 11763 | . . . . . . 7 |
4 | f1ocnv 5348 | . . . . . . 7 | |
5 | f1of 5335 | . . . . . . 7 | |
6 | 3, 4, 5 | mp2b 8 | . . . . . 6 |
7 | 6 | ffvelrni 5522 | . . . . 5 |
8 | xp2nd 6032 | . . . . 5 | |
9 | 7, 8 | syl 14 | . . . 4 |
10 | 9 | nn0zd 9129 | . . 3 |
11 | 2nn 8839 | . . . . 5 | |
12 | 11 | a1i 9 | . . . 4 |
13 | 12 | nnzd 9130 | . . 3 |
14 | dvdsmul2 11428 | . . 3 | |
15 | 10, 13, 14 | syl2anc 408 | . 2 |
16 | xp1st 6031 | . . . . . . . . . 10 | |
17 | 7, 16 | syl 14 | . . . . . . . . 9 |
18 | breq2 3903 | . . . . . . . . . . . 12 | |
19 | 18 | notbid 641 | . . . . . . . . . . 11 |
20 | 19, 1 | elrab2 2816 | . . . . . . . . . 10 |
21 | 20 | simplbi 272 | . . . . . . . . 9 |
22 | 17, 21 | syl 14 | . . . . . . . 8 |
23 | 22 | nnsqcld 10400 | . . . . . . 7 |
24 | 20 | simprbi 273 | . . . . . . . . . 10 |
25 | 17, 24 | syl 14 | . . . . . . . . 9 |
26 | 2prm 11720 | . . . . . . . . . 10 | |
27 | 22 | nnzd 9130 | . . . . . . . . . 10 |
28 | euclemma 11736 | . . . . . . . . . . 11 | |
29 | oridm 731 | . . . . . . . . . . 11 | |
30 | 28, 29 | syl6bb 195 | . . . . . . . . . 10 |
31 | 26, 27, 27, 30 | mp3an2i 1305 | . . . . . . . . 9 |
32 | 25, 31 | mtbird 647 | . . . . . . . 8 |
33 | 22 | nncnd 8698 | . . . . . . . . . 10 |
34 | 33 | sqvald 10376 | . . . . . . . . 9 |
35 | 34 | breq2d 3911 | . . . . . . . 8 |
36 | 32, 35 | mtbird 647 | . . . . . . 7 |
37 | breq2 3903 | . . . . . . . . 9 | |
38 | 37 | notbid 641 | . . . . . . . 8 |
39 | 38, 1 | elrab2 2816 | . . . . . . 7 |
40 | 23, 36, 39 | sylanbrc 413 | . . . . . 6 |
41 | 12 | nnnn0d 8988 | . . . . . . 7 |
42 | 9, 41 | nn0mulcld 8993 | . . . . . 6 |
43 | opelxp 4539 | . . . . . 6 | |
44 | 40, 42, 43 | sylanbrc 413 | . . . . 5 |
45 | 12 | nncnd 8698 | . . . . . . . . 9 |
46 | 45, 41, 9 | expmuld 10382 | . . . . . . . 8 |
47 | 46 | oveq1d 5757 | . . . . . . 7 |
48 | 12, 42 | nnexpcld 10401 | . . . . . . . . 9 |
49 | 48, 23 | nnmulcld 8733 | . . . . . . . 8 |
50 | oveq2 5750 | . . . . . . . . 9 | |
51 | oveq2 5750 | . . . . . . . . . 10 | |
52 | 51 | oveq1d 5757 | . . . . . . . . 9 |
53 | 50, 52, 2 | ovmpog 5873 | . . . . . . . 8 |
54 | 40, 42, 49, 53 | syl3anc 1201 | . . . . . . 7 |
55 | f1ocnvfv2 5647 | . . . . . . . . . . . . 13 | |
56 | 3, 55 | mpan 420 | . . . . . . . . . . . 12 |
57 | 1st2nd2 6041 | . . . . . . . . . . . . . 14 | |
58 | 7, 57 | syl 14 | . . . . . . . . . . . . 13 |
59 | 58 | fveq2d 5393 | . . . . . . . . . . . 12 |
60 | 56, 59 | eqtr3d 2152 | . . . . . . . . . . 11 |
61 | df-ov 5745 | . . . . . . . . . . 11 | |
62 | 60, 61 | syl6eqr 2168 | . . . . . . . . . 10 |
63 | 12, 9 | nnexpcld 10401 | . . . . . . . . . . . 12 |
64 | 63, 22 | nnmulcld 8733 | . . . . . . . . . . 11 |
65 | oveq2 5750 | . . . . . . . . . . . 12 | |
66 | oveq2 5750 | . . . . . . . . . . . . 13 | |
67 | 66 | oveq1d 5757 | . . . . . . . . . . . 12 |
68 | 65, 67, 2 | ovmpog 5873 | . . . . . . . . . . 11 |
69 | 17, 9, 64, 68 | syl3anc 1201 | . . . . . . . . . 10 |
70 | 62, 69 | eqtrd 2150 | . . . . . . . . 9 |
71 | 70 | oveq1d 5757 | . . . . . . . 8 |
72 | 63 | nncnd 8698 | . . . . . . . . 9 |
73 | 72, 33 | sqmuld 10391 | . . . . . . . 8 |
74 | 71, 73 | eqtrd 2150 | . . . . . . 7 |
75 | 47, 54, 74 | 3eqtr4rd 2161 | . . . . . 6 |
76 | df-ov 5745 | . . . . . 6 | |
77 | 75, 76 | syl6req 2167 | . . . . 5 |
78 | f1ocnvfv 5648 | . . . . . 6 | |
79 | 3, 78 | mpan 420 | . . . . 5 |
80 | 44, 77, 79 | sylc 62 | . . . 4 |
81 | 80 | fveq2d 5393 | . . 3 |
82 | op2ndg 6017 | . . . 4 | |
83 | 40, 42, 82 | syl2anc 408 | . . 3 |
84 | 81, 83 | eqtrd 2150 | . 2 |
85 | 15, 84 | breqtrrd 3926 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 682 w3a 947 wceq 1316 wcel 1465 crab 2397 cop 3500 class class class wbr 3899 cxp 4507 ccnv 4508 wf 5089 wf1o 5092 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 cmul 7593 cn 8684 c2 8735 cn0 8935 cz 9012 cexp 10247 cdvds 11405 cprime 11700 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-1o 6281 df-2o 6282 df-er 6397 df-en 6603 df-sup 6839 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-3 8744 df-4 8745 df-n0 8936 df-z 9013 df-uz 9283 df-q 9368 df-rp 9398 df-fz 9746 df-fzo 9875 df-fl 9998 df-mod 10051 df-seqfrec 10174 df-exp 10248 df-cj 10569 df-re 10570 df-im 10571 df-rsqrt 10725 df-abs 10726 df-dvds 11406 df-gcd 11548 df-prm 11701 |
This theorem is referenced by: sqne2sq 11766 |
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