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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 5750 | . . . . . 6 | |
2 | 1 | oveq2d 5758 | . . . . 5 |
3 | 2 | eqeq1d 2126 | . . . 4 |
4 | 3 | imbi2d 229 | . . 3 |
5 | oveq2 5750 | . . . . . 6 | |
6 | 5 | oveq2d 5758 | . . . . 5 |
7 | 6 | eqeq1d 2126 | . . . 4 |
8 | 7 | imbi2d 229 | . . 3 |
9 | oveq2 5750 | . . . . . 6 | |
10 | 9 | oveq2d 5758 | . . . . 5 |
11 | 10 | eqeq1d 2126 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | oveq2 5750 | . . . . . 6 | |
14 | 13 | oveq2d 5758 | . . . . 5 |
15 | 14 | eqeq1d 2126 | . . . 4 |
16 | 15 | imbi2d 229 | . . 3 |
17 | nncn 8696 | . . . . . 6 | |
18 | 17 | mulid1d 7751 | . . . . 5 |
19 | 18 | oveq2d 5758 | . . . 4 |
20 | nnz 9041 | . . . . . 6 | |
21 | gcdid 11601 | . . . . . 6 | |
22 | 20, 21 | syl 14 | . . . . 5 |
23 | nnre 8695 | . . . . . 6 | |
24 | nnnn0 8952 | . . . . . . 7 | |
25 | 24 | nn0ge0d 9001 | . . . . . 6 |
26 | 23, 25 | absidd 10907 | . . . . 5 |
27 | 22, 26 | eqtrd 2150 | . . . 4 |
28 | 19, 27 | eqtrd 2150 | . . 3 |
29 | 20 | adantr 274 | . . . . . . . . 9 |
30 | nnz 9041 | . . . . . . . . . 10 | |
31 | zmulcl 9075 | . . . . . . . . . 10 | |
32 | 20, 30, 31 | syl2an 287 | . . . . . . . . 9 |
33 | 1z 9048 | . . . . . . . . . 10 | |
34 | gcdaddm 11599 | . . . . . . . . . 10 | |
35 | 33, 34 | mp3an1 1287 | . . . . . . . . 9 |
36 | 29, 32, 35 | syl2anc 408 | . . . . . . . 8 |
37 | nncn 8696 | . . . . . . . . . 10 | |
38 | ax-1cn 7681 | . . . . . . . . . . . 12 | |
39 | adddi 7720 | . . . . . . . . . . . 12 | |
40 | 38, 39 | mp3an3 1289 | . . . . . . . . . . 11 |
41 | mulcom 7717 | . . . . . . . . . . . . . 14 | |
42 | 38, 41 | mpan2 421 | . . . . . . . . . . . . 13 |
43 | 42 | adantr 274 | . . . . . . . . . . . 12 |
44 | 43 | oveq2d 5758 | . . . . . . . . . . 11 |
45 | 40, 44 | eqtrd 2150 | . . . . . . . . . 10 |
46 | 17, 37, 45 | syl2an 287 | . . . . . . . . 9 |
47 | 46 | oveq2d 5758 | . . . . . . . 8 |
48 | 36, 47 | eqtr4d 2153 | . . . . . . 7 |
49 | 48 | eqeq1d 2126 | . . . . . 6 |
50 | 49 | biimpd 143 | . . . . 5 |
51 | 50 | expcom 115 | . . . 4 |
52 | 51 | a2d 26 | . . 3 |
53 | 4, 8, 12, 16, 28, 52 | nnind 8704 | . 2 |
54 | 53 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cfv 5093 (class class class)co 5742 cc 7586 c1 7589 caddc 7591 cmul 7593 cn 8688 cz 9022 cabs 10737 cgcd 11562 |
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 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fzo 9888 df-fl 10011 df-mod 10064 df-seqfrec 10187 df-exp 10261 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-dvds 11421 df-gcd 11563 |
This theorem is referenced by: gcdmultiplez 11636 rpmulgcd 11641 |
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