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Mirrors > Home > MPE Home > Th. List > gcdneg | Structured version Visualization version Unicode version |
Description: Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcdneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 6659 | . . . . 5 | |
2 | 1 | adantl 482 | . . . 4 |
3 | zcn 11382 | . . . . . . . . 9 | |
4 | 3 | negeq0d 10384 | . . . . . . . 8 |
5 | 4 | anbi2d 740 | . . . . . . 7 |
6 | 5 | adantl 482 | . . . . . 6 |
7 | oveq12 6659 | . . . . . 6 | |
8 | 6, 7 | syl6bi 243 | . . . . 5 |
9 | 8 | imp 445 | . . . 4 |
10 | 2, 9 | eqtr4d 2659 | . . 3 |
11 | gcddvds 15225 | . . . . . . 7 | |
12 | gcdcl 15228 | . . . . . . . . . 10 | |
13 | 12 | nn0zd 11480 | . . . . . . . . 9 |
14 | dvdsnegb 14999 | . . . . . . . . 9 | |
15 | 13, 14 | sylancom 701 | . . . . . . . 8 |
16 | 15 | anbi2d 740 | . . . . . . 7 |
17 | 11, 16 | mpbid 222 | . . . . . 6 |
18 | 6 | notbid 308 | . . . . . . . 8 |
19 | simpl 473 | . . . . . . . . 9 | |
20 | znegcl 11412 | . . . . . . . . . 10 | |
21 | 20 | adantl 482 | . . . . . . . . 9 |
22 | dvdslegcd 15226 | . . . . . . . . . 10 | |
23 | 22 | ex 450 | . . . . . . . . 9 |
24 | 13, 19, 21, 23 | syl3anc 1326 | . . . . . . . 8 |
25 | 18, 24 | sylbid 230 | . . . . . . 7 |
26 | 25 | com12 32 | . . . . . 6 |
27 | 17, 26 | mpdi 45 | . . . . 5 |
28 | 27 | impcom 446 | . . . 4 |
29 | gcddvds 15225 | . . . . . . . 8 | |
30 | 20, 29 | sylan2 491 | . . . . . . 7 |
31 | gcdcl 15228 | . . . . . . . . . . 11 | |
32 | 31 | nn0zd 11480 | . . . . . . . . . 10 |
33 | 20, 32 | sylan2 491 | . . . . . . . . 9 |
34 | dvdsnegb 14999 | . . . . . . . . 9 | |
35 | 33, 34 | sylancom 701 | . . . . . . . 8 |
36 | 35 | anbi2d 740 | . . . . . . 7 |
37 | 30, 36 | mpbird 247 | . . . . . 6 |
38 | simpr 477 | . . . . . . . 8 | |
39 | dvdslegcd 15226 | . . . . . . . . 9 | |
40 | 39 | ex 450 | . . . . . . . 8 |
41 | 33, 19, 38, 40 | syl3anc 1326 | . . . . . . 7 |
42 | 41 | com12 32 | . . . . . 6 |
43 | 37, 42 | mpdi 45 | . . . . 5 |
44 | 43 | impcom 446 | . . . 4 |
45 | 13 | zred 11482 | . . . . . 6 |
46 | 33 | zred 11482 | . . . . . 6 |
47 | 45, 46 | letri3d 10179 | . . . . 5 |
48 | 47 | adantr 481 | . . . 4 |
49 | 28, 44, 48 | mpbir2and 957 | . . 3 |
50 | 10, 49 | pm2.61dan 832 | . 2 |
51 | 50 | eqcomd 2628 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 (class class class)co 6650 cc0 9936 cle 10075 cneg 10267 cz 11377 cdvds 14983 cgcd 15216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 |
This theorem is referenced by: neggcd 15244 gcdabs 15250 odinv 17978 divnumden2 29564 |
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