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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 11656 | . . . . 5 | |
2 | 1 | nnsqcld 10445 | . . . 4 |
3 | 2 | nncnd 8734 | . . 3 |
4 | 3 | mulid1d 7783 | . 2 |
5 | nnsqcl 10362 | . . . . . . 7 | |
6 | 5 | nnzd 9172 | . . . . . 6 |
7 | 6 | adantr 274 | . . . . 5 |
8 | nnsqcl 10362 | . . . . . . 7 | |
9 | 8 | nnzd 9172 | . . . . . 6 |
10 | 9 | adantl 275 | . . . . 5 |
11 | nnz 9073 | . . . . . . . 8 | |
12 | nnz 9073 | . . . . . . . 8 | |
13 | gcddvds 11652 | . . . . . . . 8 | |
14 | 11, 12, 13 | syl2an 287 | . . . . . . 7 |
15 | 14 | simpld 111 | . . . . . 6 |
16 | 1 | nnzd 9172 | . . . . . . 7 |
17 | 11 | adantr 274 | . . . . . . 7 |
18 | dvdssqim 11712 | . . . . . . 7 | |
19 | 16, 17, 18 | syl2anc 408 | . . . . . 6 |
20 | 15, 19 | mpd 13 | . . . . 5 |
21 | 14 | simprd 113 | . . . . . 6 |
22 | 12 | adantl 275 | . . . . . . 7 |
23 | dvdssqim 11712 | . . . . . . 7 | |
24 | 16, 22, 23 | syl2anc 408 | . . . . . 6 |
25 | 21, 24 | mpd 13 | . . . . 5 |
26 | gcddiv 11707 | . . . . 5 | |
27 | 7, 10, 2, 20, 25, 26 | syl32anc 1224 | . . . 4 |
28 | nncn 8728 | . . . . . . 7 | |
29 | 28 | adantr 274 | . . . . . 6 |
30 | 1 | nncnd 8734 | . . . . . 6 |
31 | 1 | nnap0d 8766 | . . . . . 6 # |
32 | 29, 30, 31 | sqdivapd 10437 | . . . . 5 |
33 | nncn 8728 | . . . . . . 7 | |
34 | 33 | adantl 275 | . . . . . 6 |
35 | 34, 30, 31 | sqdivapd 10437 | . . . . 5 |
36 | 32, 35 | oveq12d 5792 | . . . 4 |
37 | gcddiv 11707 | . . . . . . 7 | |
38 | 17, 22, 1, 14, 37 | syl31anc 1219 | . . . . . 6 |
39 | 30, 31 | dividapd 8546 | . . . . . 6 |
40 | 38, 39 | eqtr3d 2174 | . . . . 5 |
41 | 1 | nnne0d 8765 | . . . . . . . . 9 |
42 | dvdsval2 11496 | . . . . . . . . 9 | |
43 | 16, 41, 17, 42 | syl3anc 1216 | . . . . . . . 8 |
44 | 15, 43 | mpbid 146 | . . . . . . 7 |
45 | nnre 8727 | . . . . . . . . 9 | |
46 | 45 | adantr 274 | . . . . . . . 8 |
47 | 1 | nnred 8733 | . . . . . . . 8 |
48 | nngt0 8745 | . . . . . . . . 9 | |
49 | 48 | adantr 274 | . . . . . . . 8 |
50 | 1 | nngt0d 8764 | . . . . . . . 8 |
51 | 46, 47, 49, 50 | divgt0d 8693 | . . . . . . 7 |
52 | elnnz 9064 | . . . . . . 7 | |
53 | 44, 51, 52 | sylanbrc 413 | . . . . . 6 |
54 | dvdsval2 11496 | . . . . . . . . 9 | |
55 | 16, 41, 22, 54 | syl3anc 1216 | . . . . . . . 8 |
56 | 21, 55 | mpbid 146 | . . . . . . 7 |
57 | nnre 8727 | . . . . . . . . 9 | |
58 | 57 | adantl 275 | . . . . . . . 8 |
59 | nngt0 8745 | . . . . . . . . 9 | |
60 | 59 | adantl 275 | . . . . . . . 8 |
61 | 58, 47, 60, 50 | divgt0d 8693 | . . . . . . 7 |
62 | elnnz 9064 | . . . . . . 7 | |
63 | 56, 61, 62 | sylanbrc 413 | . . . . . 6 |
64 | 2nn 8881 | . . . . . . 7 | |
65 | rppwr 11716 | . . . . . . 7 | |
66 | 64, 65 | mp3an3 1304 | . . . . . 6 |
67 | 53, 63, 66 | syl2anc 408 | . . . . 5 |
68 | 40, 67 | mpd 13 | . . . 4 |
69 | 27, 36, 68 | 3eqtr2d 2178 | . . 3 |
70 | 6, 9 | anim12i 336 | . . . . . 6 |
71 | 5 | nnne0d 8765 | . . . . . . . . 9 |
72 | 71 | neneqd 2329 | . . . . . . . 8 |
73 | 72 | intnanrd 917 | . . . . . . 7 |
74 | 73 | adantr 274 | . . . . . 6 |
75 | gcdn0cl 11651 | . . . . . 6 | |
76 | 70, 74, 75 | syl2anc 408 | . . . . 5 |
77 | 76 | nncnd 8734 | . . . 4 |
78 | 2 | nnap0d 8766 | . . . 4 # |
79 | ax-1cn 7713 | . . . . 5 | |
80 | divmulap 8435 | . . . . 5 # | |
81 | 79, 80 | mp3an2 1303 | . . . 4 # |
82 | 77, 3, 78, 81 | syl12anc 1214 | . . 3 |
83 | 69, 82 | mpbid 146 | . 2 |
84 | 4, 83 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 cmul 7625 clt 7800 # cap 8343 cdiv 8432 cn 8720 c2 8771 cz 9054 cexp 10292 cdvds 11493 cgcd 11635 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-dvds 11494 df-gcd 11636 |
This theorem is referenced by: dvdssqlem 11718 nn0gcdsq 11878 |
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