Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nn0gcdsq | Unicode version |
Description: Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
nn0gcdsq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8972 | . 2 | |
2 | elnn0 8972 | . 2 | |
3 | sqgcd 11706 | . . 3 | |
4 | nncn 8721 | . . . . . . 7 | |
5 | abssq 10846 | . . . . . . 7 | |
6 | 4, 5 | syl 14 | . . . . . 6 |
7 | nnz 9066 | . . . . . . . 8 | |
8 | gcd0id 11656 | . . . . . . . 8 | |
9 | 7, 8 | syl 14 | . . . . . . 7 |
10 | 9 | oveq1d 5782 | . . . . . 6 |
11 | sq0 10376 | . . . . . . . . 9 | |
12 | 11 | a1i 9 | . . . . . . . 8 |
13 | 12 | oveq1d 5782 | . . . . . . 7 |
14 | zsqcl 10356 | . . . . . . . 8 | |
15 | gcd0id 11656 | . . . . . . . 8 | |
16 | 7, 14, 15 | 3syl 17 | . . . . . . 7 |
17 | 13, 16 | eqtrd 2170 | . . . . . 6 |
18 | 6, 10, 17 | 3eqtr4d 2180 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 |
20 | oveq1 5774 | . . . . . . 7 | |
21 | 20 | oveq1d 5782 | . . . . . 6 |
22 | oveq1 5774 | . . . . . . 7 | |
23 | 22 | oveq1d 5782 | . . . . . 6 |
24 | 21, 23 | eqeq12d 2152 | . . . . 5 |
25 | 24 | adantr 274 | . . . 4 |
26 | 19, 25 | mpbird 166 | . . 3 |
27 | nncn 8721 | . . . . . . 7 | |
28 | abssq 10846 | . . . . . . 7 | |
29 | 27, 28 | syl 14 | . . . . . 6 |
30 | nnz 9066 | . . . . . . . 8 | |
31 | gcdid0 11657 | . . . . . . . 8 | |
32 | 30, 31 | syl 14 | . . . . . . 7 |
33 | 32 | oveq1d 5782 | . . . . . 6 |
34 | 11 | a1i 9 | . . . . . . . 8 |
35 | 34 | oveq2d 5783 | . . . . . . 7 |
36 | zsqcl 10356 | . . . . . . . 8 | |
37 | gcdid0 11657 | . . . . . . . 8 | |
38 | 30, 36, 37 | 3syl 17 | . . . . . . 7 |
39 | 35, 38 | eqtrd 2170 | . . . . . 6 |
40 | 29, 33, 39 | 3eqtr4d 2180 | . . . . 5 |
41 | 40 | adantr 274 | . . . 4 |
42 | oveq2 5775 | . . . . . . 7 | |
43 | 42 | oveq1d 5782 | . . . . . 6 |
44 | oveq1 5774 | . . . . . . 7 | |
45 | 44 | oveq2d 5783 | . . . . . 6 |
46 | 43, 45 | eqeq12d 2152 | . . . . 5 |
47 | 46 | adantl 275 | . . . 4 |
48 | 41, 47 | mpbird 166 | . . 3 |
49 | gcd0val 11638 | . . . . . 6 | |
50 | 49 | oveq1i 5777 | . . . . 5 |
51 | 11, 11 | oveq12i 5779 | . . . . . 6 |
52 | 51, 49 | eqtri 2158 | . . . . 5 |
53 | 11, 50, 52 | 3eqtr4i 2168 | . . . 4 |
54 | oveq12 5776 | . . . . 5 | |
55 | 54 | oveq1d 5782 | . . . 4 |
56 | 22, 44 | oveqan12d 5786 | . . . 4 |
57 | 53, 55, 56 | 3eqtr4a 2196 | . . 3 |
58 | 3, 26, 48, 57 | ccase 948 | . 2 |
59 | 1, 2, 58 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cfv 5118 (class class class)co 5767 cc 7611 cc0 7613 cn 8713 c2 8764 cn0 8970 cz 9047 cexp 10285 cabs 10762 cgcd 11624 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-fl 10036 df-mod 10089 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-dvds 11483 df-gcd 11625 |
This theorem is referenced by: zgcdsq 11868 |
Copyright terms: Public domain | W3C validator |