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Mirrors > Home > ILE Home > Th. List > gcdaddm | Unicode version |
Description: Adding a multiple of one operand of the operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.) |
Ref | Expression |
---|---|
gcdaddm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcddvds 11652 | . . . . . . . . 9 | |
2 | 1 | 3adant1 999 | . . . . . . . 8 |
3 | 2 | simpld 111 | . . . . . . 7 |
4 | simp1 981 | . . . . . . . . . 10 | |
5 | 1zzd 9081 | . . . . . . . . . 10 | |
6 | gcdcl 11655 | . . . . . . . . . . . 12 | |
7 | 6 | 3adant1 999 | . . . . . . . . . . 11 |
8 | 7 | nn0zd 9171 | . . . . . . . . . 10 |
9 | simp2 982 | . . . . . . . . . 10 | |
10 | simp3 983 | . . . . . . . . . 10 | |
11 | dvds2ln 11526 | . . . . . . . . . 10 | |
12 | 4, 5, 8, 9, 10, 11 | syl23anc 1223 | . . . . . . . . 9 |
13 | 2, 12 | mpd 13 | . . . . . . . 8 |
14 | 10 | zcnd 9174 | . . . . . . . . . 10 |
15 | 14 | mulid2d 7784 | . . . . . . . . 9 |
16 | 15 | oveq2d 5790 | . . . . . . . 8 |
17 | 13, 16 | breqtrd 3954 | . . . . . . 7 |
18 | 3, 17 | jca 304 | . . . . . 6 |
19 | 4, 9 | zmulcld 9179 | . . . . . . . 8 |
20 | 19, 10 | zaddcld 9177 | . . . . . . 7 |
21 | dvdslegcd 11653 | . . . . . . . 8 | |
22 | 21 | ex 114 | . . . . . . 7 |
23 | 8, 9, 20, 22 | syl3anc 1216 | . . . . . 6 |
24 | 18, 23 | mpid 42 | . . . . 5 |
25 | gcddvds 11652 | . . . . . . . . 9 | |
26 | 9, 20, 25 | syl2anc 408 | . . . . . . . 8 |
27 | 26 | simpld 111 | . . . . . . 7 |
28 | 4 | znegcld 9175 | . . . . . . . . . 10 |
29 | 9, 20 | gcdcld 11657 | . . . . . . . . . . 11 |
30 | 29 | nn0zd 9171 | . . . . . . . . . 10 |
31 | dvds2ln 11526 | . . . . . . . . . 10 | |
32 | 28, 5, 30, 9, 20, 31 | syl23anc 1223 | . . . . . . . . 9 |
33 | 26, 32 | mpd 13 | . . . . . . . 8 |
34 | 4 | zcnd 9174 | . . . . . . . . . . 11 |
35 | 9 | zcnd 9174 | . . . . . . . . . . 11 |
36 | 34, 35 | mulneg1d 8173 | . . . . . . . . . 10 |
37 | 20 | zcnd 9174 | . . . . . . . . . . 11 |
38 | 37 | mulid2d 7784 | . . . . . . . . . 10 |
39 | 36, 38 | oveq12d 5792 | . . . . . . . . 9 |
40 | 34, 35 | mulcld 7786 | . . . . . . . . . . . . 13 |
41 | 40 | negcld 8060 | . . . . . . . . . . . . 13 |
42 | 40, 41 | addcomd 7913 | . . . . . . . . . . . 12 |
43 | 40 | negidd 8063 | . . . . . . . . . . . 12 |
44 | 42, 43 | eqtr3d 2174 | . . . . . . . . . . 11 |
45 | 44 | oveq1d 5789 | . . . . . . . . . 10 |
46 | 41, 40, 14 | addassd 7788 | . . . . . . . . . 10 |
47 | 14 | addid2d 7912 | . . . . . . . . . 10 |
48 | 45, 46, 47 | 3eqtr3d 2180 | . . . . . . . . 9 |
49 | 39, 48 | eqtrd 2172 | . . . . . . . 8 |
50 | 33, 49 | breqtrd 3954 | . . . . . . 7 |
51 | 27, 50 | jca 304 | . . . . . 6 |
52 | dvdslegcd 11653 | . . . . . . . 8 | |
53 | 52 | ex 114 | . . . . . . 7 |
54 | 30, 9, 10, 53 | syl3anc 1216 | . . . . . 6 |
55 | 51, 54 | mpid 42 | . . . . 5 |
56 | 24, 55 | anim12d 333 | . . . 4 |
57 | 7 | nn0red 9031 | . . . . 5 |
58 | 29 | nn0red 9031 | . . . . 5 |
59 | 57, 58 | letri3d 7879 | . . . 4 |
60 | 56, 59 | sylibrd 168 | . . 3 |
61 | 0zd 9066 | . . . . . . 7 | |
62 | zdceq 9126 | . . . . . . 7 DECID | |
63 | 9, 61, 62 | syl2anc 408 | . . . . . 6 DECID |
64 | zdceq 9126 | . . . . . . 7 DECID | |
65 | 20, 61, 64 | syl2anc 408 | . . . . . 6 DECID |
66 | dcan 918 | . . . . . 6 DECID DECID DECID | |
67 | 63, 65, 66 | sylc 62 | . . . . 5 DECID |
68 | zdceq 9126 | . . . . . . 7 DECID | |
69 | 10, 61, 68 | syl2anc 408 | . . . . . 6 DECID |
70 | dcan 918 | . . . . . 6 DECID DECID DECID | |
71 | 63, 69, 70 | sylc 62 | . . . . 5 DECID |
72 | orandc 923 | . . . . 5 DECID DECID | |
73 | 67, 71, 72 | syl2anc 408 | . . . 4 |
74 | simpr 109 | . . . . . . . . . . . 12 | |
75 | 74 | oveq2d 5790 | . . . . . . . . . . 11 |
76 | 34 | mul01d 8155 | . . . . . . . . . . . 12 |
77 | 76 | adantr 274 | . . . . . . . . . . 11 |
78 | 75, 77 | eqtrd 2172 | . . . . . . . . . 10 |
79 | 78 | oveq1d 5789 | . . . . . . . . 9 |
80 | 47 | adantr 274 | . . . . . . . . 9 |
81 | 79, 80 | eqtrd 2172 | . . . . . . . 8 |
82 | 81 | eqeq1d 2148 | . . . . . . 7 |
83 | 82 | pm5.32da 447 | . . . . . 6 |
84 | oveq12 5783 | . . . . . . . . 9 | |
85 | 84 | adantl 275 | . . . . . . . 8 |
86 | oveq12 5783 | . . . . . . . . . 10 | |
87 | 83, 86 | syl6bir 163 | . . . . . . . . 9 |
88 | 87 | imp 123 | . . . . . . . 8 |
89 | 85, 88 | eqtr4d 2175 | . . . . . . 7 |
90 | 89 | ex 114 | . . . . . 6 |
91 | 83, 90 | sylbid 149 | . . . . 5 |
92 | 91, 90 | jaod 706 | . . . 4 |
93 | 73, 92 | sylbird 169 | . . 3 |
94 | dcn 827 | . . . . . 6 DECID DECID | |
95 | 67, 94 | syl 14 | . . . . 5 DECID |
96 | dcn 827 | . . . . . 6 DECID DECID | |
97 | 71, 96 | syl 14 | . . . . 5 DECID |
98 | dcan 918 | . . . . 5 DECID DECID DECID | |
99 | 95, 97, 98 | sylc 62 | . . . 4 DECID |
100 | exmiddc 821 | . . . 4 DECID | |
101 | 99, 100 | syl 14 | . . 3 |
102 | 60, 93, 101 | mpjaod 707 | . 2 |
103 | 40, 14 | addcomd 7913 | . . 3 |
104 | 103 | oveq2d 5790 | . 2 |
105 | 102, 104 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc0 7620 c1 7621 caddc 7623 cmul 7625 cle 7801 cneg 7934 cn0 8977 cz 9054 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: gcdadd 11673 gcdid 11674 modgcd 11679 gcdmultipled 11681 gcdmultiple 11708 |
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