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Mirrors > Home > ILE Home > Th. List > sqgcd | Unicode version |
Description: Square distributes over gcd. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
sqgcd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdnncl 11935 | . . . . 5 | |
2 | 1 | nnsqcld 10644 | . . . 4 |
3 | 2 | nncnd 8906 | . . 3 |
4 | 3 | mulid1d 7949 | . 2 |
5 | nnsqcl 10559 | . . . . . . 7 | |
6 | 5 | nnzd 9347 | . . . . . 6 |
7 | 6 | adantr 276 | . . . . 5 |
8 | nnsqcl 10559 | . . . . . . 7 | |
9 | 8 | nnzd 9347 | . . . . . 6 |
10 | 9 | adantl 277 | . . . . 5 |
11 | nnz 9245 | . . . . . . . 8 | |
12 | nnz 9245 | . . . . . . . 8 | |
13 | gcddvds 11931 | . . . . . . . 8 | |
14 | 11, 12, 13 | syl2an 289 | . . . . . . 7 |
15 | 14 | simpld 112 | . . . . . 6 |
16 | 1 | nnzd 9347 | . . . . . . 7 |
17 | 11 | adantr 276 | . . . . . . 7 |
18 | dvdssqim 11992 | . . . . . . 7 | |
19 | 16, 17, 18 | syl2anc 411 | . . . . . 6 |
20 | 15, 19 | mpd 13 | . . . . 5 |
21 | 14 | simprd 114 | . . . . . 6 |
22 | 12 | adantl 277 | . . . . . . 7 |
23 | dvdssqim 11992 | . . . . . . 7 | |
24 | 16, 22, 23 | syl2anc 411 | . . . . . 6 |
25 | 21, 24 | mpd 13 | . . . . 5 |
26 | gcddiv 11987 | . . . . 5 | |
27 | 7, 10, 2, 20, 25, 26 | syl32anc 1246 | . . . 4 |
28 | nncn 8900 | . . . . . . 7 | |
29 | 28 | adantr 276 | . . . . . 6 |
30 | 1 | nncnd 8906 | . . . . . 6 |
31 | 1 | nnap0d 8938 | . . . . . 6 # |
32 | 29, 30, 31 | sqdivapd 10636 | . . . . 5 |
33 | nncn 8900 | . . . . . . 7 | |
34 | 33 | adantl 277 | . . . . . 6 |
35 | 34, 30, 31 | sqdivapd 10636 | . . . . 5 |
36 | 32, 35 | oveq12d 5883 | . . . 4 |
37 | gcddiv 11987 | . . . . . . 7 | |
38 | 17, 22, 1, 14, 37 | syl31anc 1241 | . . . . . 6 |
39 | 30, 31 | dividapd 8716 | . . . . . 6 |
40 | 38, 39 | eqtr3d 2210 | . . . . 5 |
41 | 1 | nnne0d 8937 | . . . . . . . . 9 |
42 | dvdsval2 11765 | . . . . . . . . 9 | |
43 | 16, 41, 17, 42 | syl3anc 1238 | . . . . . . . 8 |
44 | 15, 43 | mpbid 147 | . . . . . . 7 |
45 | nnre 8899 | . . . . . . . . 9 | |
46 | 45 | adantr 276 | . . . . . . . 8 |
47 | 1 | nnred 8905 | . . . . . . . 8 |
48 | nngt0 8917 | . . . . . . . . 9 | |
49 | 48 | adantr 276 | . . . . . . . 8 |
50 | 1 | nngt0d 8936 | . . . . . . . 8 |
51 | 46, 47, 49, 50 | divgt0d 8865 | . . . . . . 7 |
52 | elnnz 9236 | . . . . . . 7 | |
53 | 44, 51, 52 | sylanbrc 417 | . . . . . 6 |
54 | dvdsval2 11765 | . . . . . . . . 9 | |
55 | 16, 41, 22, 54 | syl3anc 1238 | . . . . . . . 8 |
56 | 21, 55 | mpbid 147 | . . . . . . 7 |
57 | nnre 8899 | . . . . . . . . 9 | |
58 | 57 | adantl 277 | . . . . . . . 8 |
59 | nngt0 8917 | . . . . . . . . 9 | |
60 | 59 | adantl 277 | . . . . . . . 8 |
61 | 58, 47, 60, 50 | divgt0d 8865 | . . . . . . 7 |
62 | elnnz 9236 | . . . . . . 7 | |
63 | 56, 61, 62 | sylanbrc 417 | . . . . . 6 |
64 | 2nn 9053 | . . . . . . 7 | |
65 | rppwr 11996 | . . . . . . 7 | |
66 | 64, 65 | mp3an3 1326 | . . . . . 6 |
67 | 53, 63, 66 | syl2anc 411 | . . . . 5 |
68 | 40, 67 | mpd 13 | . . . 4 |
69 | 27, 36, 68 | 3eqtr2d 2214 | . . 3 |
70 | 6, 9 | anim12i 338 | . . . . . 6 |
71 | 5 | nnne0d 8937 | . . . . . . . . 9 |
72 | 71 | neneqd 2366 | . . . . . . . 8 |
73 | 72 | intnanrd 932 | . . . . . . 7 |
74 | 73 | adantr 276 | . . . . . 6 |
75 | gcdn0cl 11930 | . . . . . 6 | |
76 | 70, 74, 75 | syl2anc 411 | . . . . 5 |
77 | 76 | nncnd 8906 | . . . 4 |
78 | 2 | nnap0d 8938 | . . . 4 # |
79 | ax-1cn 7879 | . . . . 5 | |
80 | divmulap 8605 | . . . . 5 # | |
81 | 79, 80 | mp3an2 1325 | . . . 4 # |
82 | 77, 3, 78, 81 | syl12anc 1236 | . . 3 |
83 | 69, 82 | mpbid 147 | . 2 |
84 | 4, 83 | eqtr3d 2210 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wne 2345 class class class wbr 3998 (class class class)co 5865 cc 7784 cr 7785 cc0 7786 c1 7787 cmul 7791 clt 7966 # cap 8512 cdiv 8602 cn 8892 c2 8943 cz 9226 cexp 10489 cdvds 11762 cgcd 11910 |
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 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-stab 831 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-sup 6973 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-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-fz 9980 df-fzo 10113 df-fl 10240 df-mod 10293 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-dvds 11763 df-gcd 11911 |
This theorem is referenced by: dvdssqlem 11998 nn0gcdsq 12167 pythagtriplem3 12234 |
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