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Mirrors > Home > ILE Home > Th. List > gcdmultiple | Unicode version |
Description: The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
gcdmultiple |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5775 | . . . . . 6 | |
2 | 1 | oveq2d 5783 | . . . . 5 |
3 | 2 | eqeq1d 2146 | . . . 4 |
4 | 3 | imbi2d 229 | . . 3 |
5 | oveq2 5775 | . . . . . 6 | |
6 | 5 | oveq2d 5783 | . . . . 5 |
7 | 6 | eqeq1d 2146 | . . . 4 |
8 | 7 | imbi2d 229 | . . 3 |
9 | oveq2 5775 | . . . . . 6 | |
10 | 9 | oveq2d 5783 | . . . . 5 |
11 | 10 | eqeq1d 2146 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | oveq2 5775 | . . . . . 6 | |
14 | 13 | oveq2d 5783 | . . . . 5 |
15 | 14 | eqeq1d 2146 | . . . 4 |
16 | 15 | imbi2d 229 | . . 3 |
17 | nncn 8721 | . . . . . 6 | |
18 | 17 | mulid1d 7776 | . . . . 5 |
19 | 18 | oveq2d 5783 | . . . 4 |
20 | nnz 9066 | . . . . . 6 | |
21 | gcdid 11663 | . . . . . 6 | |
22 | 20, 21 | syl 14 | . . . . 5 |
23 | nnre 8720 | . . . . . 6 | |
24 | nnnn0 8977 | . . . . . . 7 | |
25 | 24 | nn0ge0d 9026 | . . . . . 6 |
26 | 23, 25 | absidd 10932 | . . . . 5 |
27 | 22, 26 | eqtrd 2170 | . . . 4 |
28 | 19, 27 | eqtrd 2170 | . . 3 |
29 | 20 | adantr 274 | . . . . . . . . 9 |
30 | nnz 9066 | . . . . . . . . . 10 | |
31 | zmulcl 9100 | . . . . . . . . . 10 | |
32 | 20, 30, 31 | syl2an 287 | . . . . . . . . 9 |
33 | 1z 9073 | . . . . . . . . . 10 | |
34 | gcdaddm 11661 | . . . . . . . . . 10 | |
35 | 33, 34 | mp3an1 1302 | . . . . . . . . 9 |
36 | 29, 32, 35 | syl2anc 408 | . . . . . . . 8 |
37 | nncn 8721 | . . . . . . . . . 10 | |
38 | ax-1cn 7706 | . . . . . . . . . . . 12 | |
39 | adddi 7745 | . . . . . . . . . . . 12 | |
40 | 38, 39 | mp3an3 1304 | . . . . . . . . . . 11 |
41 | mulcom 7742 | . . . . . . . . . . . . . 14 | |
42 | 38, 41 | mpan2 421 | . . . . . . . . . . . . 13 |
43 | 42 | adantr 274 | . . . . . . . . . . . 12 |
44 | 43 | oveq2d 5783 | . . . . . . . . . . 11 |
45 | 40, 44 | eqtrd 2170 | . . . . . . . . . 10 |
46 | 17, 37, 45 | syl2an 287 | . . . . . . . . 9 |
47 | 46 | oveq2d 5783 | . . . . . . . 8 |
48 | 36, 47 | eqtr4d 2173 | . . . . . . 7 |
49 | 48 | eqeq1d 2146 | . . . . . 6 |
50 | 49 | biimpd 143 | . . . . 5 |
51 | 50 | expcom 115 | . . . 4 |
52 | 51 | a2d 26 | . . 3 |
53 | 4, 8, 12, 16, 28, 52 | nnind 8729 | . 2 |
54 | 53 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cfv 5118 (class class class)co 5767 cc 7611 c1 7614 caddc 7616 cmul 7618 cn 8713 cz 9047 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: gcdmultiplez 11698 rpmulgcd 11703 |
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