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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 11568 | . . . . 5 | |
2 | 1 | nnsqcld 10400 | . . . 4 |
3 | 2 | nncnd 8698 | . . 3 |
4 | 3 | mulid1d 7751 | . 2 |
5 | nnsqcl 10317 | . . . . . . 7 | |
6 | 5 | nnzd 9130 | . . . . . 6 |
7 | 6 | adantr 274 | . . . . 5 |
8 | nnsqcl 10317 | . . . . . . 7 | |
9 | 8 | nnzd 9130 | . . . . . 6 |
10 | 9 | adantl 275 | . . . . 5 |
11 | nnz 9031 | . . . . . . . 8 | |
12 | nnz 9031 | . . . . . . . 8 | |
13 | gcddvds 11564 | . . . . . . . 8 | |
14 | 11, 12, 13 | syl2an 287 | . . . . . . 7 |
15 | 14 | simpld 111 | . . . . . 6 |
16 | 1 | nnzd 9130 | . . . . . . 7 |
17 | 11 | adantr 274 | . . . . . . 7 |
18 | dvdssqim 11624 | . . . . . . 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 11624 | . . . . . . 7 | |
24 | 16, 22, 23 | syl2anc 408 | . . . . . 6 |
25 | 21, 24 | mpd 13 | . . . . 5 |
26 | gcddiv 11619 | . . . . 5 | |
27 | 7, 10, 2, 20, 25, 26 | syl32anc 1209 | . . . 4 |
28 | nncn 8692 | . . . . . . 7 | |
29 | 28 | adantr 274 | . . . . . 6 |
30 | 1 | nncnd 8698 | . . . . . 6 |
31 | 1 | nnap0d 8730 | . . . . . 6 # |
32 | 29, 30, 31 | sqdivapd 10392 | . . . . 5 |
33 | nncn 8692 | . . . . . . 7 | |
34 | 33 | adantl 275 | . . . . . 6 |
35 | 34, 30, 31 | sqdivapd 10392 | . . . . 5 |
36 | 32, 35 | oveq12d 5760 | . . . 4 |
37 | gcddiv 11619 | . . . . . . 7 | |
38 | 17, 22, 1, 14, 37 | syl31anc 1204 | . . . . . 6 |
39 | 30, 31 | dividapd 8513 | . . . . . 6 |
40 | 38, 39 | eqtr3d 2152 | . . . . 5 |
41 | 1 | nnne0d 8729 | . . . . . . . . 9 |
42 | dvdsval2 11408 | . . . . . . . . 9 | |
43 | 16, 41, 17, 42 | syl3anc 1201 | . . . . . . . 8 |
44 | 15, 43 | mpbid 146 | . . . . . . 7 |
45 | nnre 8691 | . . . . . . . . 9 | |
46 | 45 | adantr 274 | . . . . . . . 8 |
47 | 1 | nnred 8697 | . . . . . . . 8 |
48 | nngt0 8709 | . . . . . . . . 9 | |
49 | 48 | adantr 274 | . . . . . . . 8 |
50 | 1 | nngt0d 8728 | . . . . . . . 8 |
51 | 46, 47, 49, 50 | divgt0d 8657 | . . . . . . 7 |
52 | elnnz 9022 | . . . . . . 7 | |
53 | 44, 51, 52 | sylanbrc 413 | . . . . . 6 |
54 | dvdsval2 11408 | . . . . . . . . 9 | |
55 | 16, 41, 22, 54 | syl3anc 1201 | . . . . . . . 8 |
56 | 21, 55 | mpbid 146 | . . . . . . 7 |
57 | nnre 8691 | . . . . . . . . 9 | |
58 | 57 | adantl 275 | . . . . . . . 8 |
59 | nngt0 8709 | . . . . . . . . 9 | |
60 | 59 | adantl 275 | . . . . . . . 8 |
61 | 58, 47, 60, 50 | divgt0d 8657 | . . . . . . 7 |
62 | elnnz 9022 | . . . . . . 7 | |
63 | 56, 61, 62 | sylanbrc 413 | . . . . . 6 |
64 | 2nn 8839 | . . . . . . 7 | |
65 | rppwr 11628 | . . . . . . 7 | |
66 | 64, 65 | mp3an3 1289 | . . . . . 6 |
67 | 53, 63, 66 | syl2anc 408 | . . . . 5 |
68 | 40, 67 | mpd 13 | . . . 4 |
69 | 27, 36, 68 | 3eqtr2d 2156 | . . 3 |
70 | 6, 9 | anim12i 336 | . . . . . 6 |
71 | 5 | nnne0d 8729 | . . . . . . . . 9 |
72 | 71 | neneqd 2306 | . . . . . . . 8 |
73 | 72 | intnanrd 902 | . . . . . . 7 |
74 | 73 | adantr 274 | . . . . . 6 |
75 | gcdn0cl 11563 | . . . . . 6 | |
76 | 70, 74, 75 | syl2anc 408 | . . . . 5 |
77 | 76 | nncnd 8698 | . . . 4 |
78 | 2 | nnap0d 8730 | . . . 4 # |
79 | ax-1cn 7681 | . . . . 5 | |
80 | divmulap 8402 | . . . . 5 # | |
81 | 79, 80 | mp3an2 1288 | . . . 4 # |
82 | 77, 3, 78, 81 | syl12anc 1199 | . . 3 |
83 | 69, 82 | mpbid 146 | . 2 |
84 | 4, 83 | eqtr3d 2152 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wne 2285 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 cmul 7593 clt 7768 # cap 8310 cdiv 8399 cn 8684 c2 8735 cz 9012 cexp 10247 cdvds 11405 cgcd 11547 |
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-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 |
This theorem is referenced by: dvdssqlem 11630 nn0gcdsq 11789 |
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