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Mirrors > Home > MPE Home > Th. List > mumul | Structured version Visualization version Unicode version |
Description: The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
mumul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1065 | . . . . . 6 | |
2 | mucl 24867 | . . . . . 6 | |
3 | 1, 2 | syl 17 | . . . . 5 |
4 | 3 | zcnd 11483 | . . . 4 |
5 | 4 | mul02d 10234 | . . 3 |
6 | simpr 477 | . . . 4 | |
7 | 6 | oveq1d 6665 | . . 3 |
8 | mumullem1 24905 | . . . 4 | |
9 | 8 | 3adantl3 1219 | . . 3 |
10 | 5, 7, 9 | 3eqtr4rd 2667 | . 2 |
11 | simpl1 1064 | . . . . . 6 | |
12 | mucl 24867 | . . . . . 6 | |
13 | 11, 12 | syl 17 | . . . . 5 |
14 | 13 | zcnd 11483 | . . . 4 |
15 | 14 | mul01d 10235 | . . 3 |
16 | simpr 477 | . . . 4 | |
17 | 16 | oveq2d 6666 | . . 3 |
18 | nncn 11028 | . . . . . . . 8 | |
19 | nncn 11028 | . . . . . . . 8 | |
20 | mulcom 10022 | . . . . . . . 8 | |
21 | 18, 19, 20 | syl2an 494 | . . . . . . 7 |
22 | 21 | fveq2d 6195 | . . . . . 6 |
23 | 22 | adantr 481 | . . . . 5 |
24 | mumullem1 24905 | . . . . . 6 | |
25 | 24 | ancom1s 847 | . . . . 5 |
26 | 23, 25 | eqtrd 2656 | . . . 4 |
27 | 26 | 3adantl3 1219 | . . 3 |
28 | 15, 17, 27 | 3eqtr4rd 2667 | . 2 |
29 | simpl1 1064 | . . . . 5 | |
30 | simpl2 1065 | . . . . 5 | |
31 | 29, 30 | nnmulcld 11068 | . . . 4 |
32 | mumullem2 24906 | . . . 4 | |
33 | muval2 24860 | . . . 4 | |
34 | 31, 32, 33 | syl2anc 693 | . . 3 |
35 | neg1cn 11124 | . . . . . 6 | |
36 | 35 | a1i 11 | . . . . 5 |
37 | fzfi 12771 | . . . . . . 7 | |
38 | prmnn 15388 | . . . . . . . . . 10 | |
39 | 38 | ssriv 3607 | . . . . . . . . 9 |
40 | rabss2 3685 | . . . . . . . . 9 | |
41 | 39, 40 | ax-mp 5 | . . . . . . . 8 |
42 | dvdsssfz1 15040 | . . . . . . . . 9 | |
43 | 30, 42 | syl 17 | . . . . . . . 8 |
44 | 41, 43 | syl5ss 3614 | . . . . . . 7 |
45 | ssfi 8180 | . . . . . . 7 | |
46 | 37, 44, 45 | sylancr 695 | . . . . . 6 |
47 | hashcl 13147 | . . . . . 6 | |
48 | 46, 47 | syl 17 | . . . . 5 |
49 | fzfi 12771 | . . . . . . 7 | |
50 | rabss2 3685 | . . . . . . . . 9 | |
51 | 39, 50 | ax-mp 5 | . . . . . . . 8 |
52 | dvdsssfz1 15040 | . . . . . . . . 9 | |
53 | 29, 52 | syl 17 | . . . . . . . 8 |
54 | 51, 53 | syl5ss 3614 | . . . . . . 7 |
55 | ssfi 8180 | . . . . . . 7 | |
56 | 49, 54, 55 | sylancr 695 | . . . . . 6 |
57 | hashcl 13147 | . . . . . 6 | |
58 | 56, 57 | syl 17 | . . . . 5 |
59 | 36, 48, 58 | expaddd 13010 | . . . 4 |
60 | simpr 477 | . . . . . . . . . 10 | |
61 | simpl1 1064 | . . . . . . . . . . . 12 | |
62 | 61 | nnzd 11481 | . . . . . . . . . . 11 |
63 | 62 | adantlr 751 | . . . . . . . . . 10 |
64 | simpl2 1065 | . . . . . . . . . . . 12 | |
65 | 64 | nnzd 11481 | . . . . . . . . . . 11 |
66 | 65 | adantlr 751 | . . . . . . . . . 10 |
67 | euclemma 15425 | . . . . . . . . . 10 | |
68 | 60, 63, 66, 67 | syl3anc 1326 | . . . . . . . . 9 |
69 | 68 | rabbidva 3188 | . . . . . . . 8 |
70 | unrab 3898 | . . . . . . . 8 | |
71 | 69, 70 | syl6eqr 2674 | . . . . . . 7 |
72 | 71 | fveq2d 6195 | . . . . . 6 |
73 | inrab 3899 | . . . . . . . 8 | |
74 | nprmdvds1 15418 | . . . . . . . . . . . 12 | |
75 | 74 | adantl 482 | . . . . . . . . . . 11 |
76 | prmz 15389 | . . . . . . . . . . . . . 14 | |
77 | 76 | adantl 482 | . . . . . . . . . . . . 13 |
78 | dvdsgcd 15261 | . . . . . . . . . . . . 13 | |
79 | 77, 63, 66, 78 | syl3anc 1326 | . . . . . . . . . . . 12 |
80 | simpll3 1102 | . . . . . . . . . . . . 13 | |
81 | 80 | breq2d 4665 | . . . . . . . . . . . 12 |
82 | 79, 81 | sylibd 229 | . . . . . . . . . . 11 |
83 | 75, 82 | mtod 189 | . . . . . . . . . 10 |
84 | 83 | ralrimiva 2966 | . . . . . . . . 9 |
85 | rabeq0 3957 | . . . . . . . . 9 | |
86 | 84, 85 | sylibr 224 | . . . . . . . 8 |
87 | 73, 86 | syl5eq 2668 | . . . . . . 7 |
88 | hashun 13171 | . . . . . . 7 | |
89 | 56, 46, 87, 88 | syl3anc 1326 | . . . . . 6 |
90 | 72, 89 | eqtrd 2656 | . . . . 5 |
91 | 90 | oveq2d 6666 | . . . 4 |
92 | simprl 794 | . . . . . 6 | |
93 | muval2 24860 | . . . . . 6 | |
94 | 29, 92, 93 | syl2anc 693 | . . . . 5 |
95 | simprr 796 | . . . . . 6 | |
96 | muval2 24860 | . . . . . 6 | |
97 | 30, 95, 96 | syl2anc 693 | . . . . 5 |
98 | 94, 97 | oveq12d 6668 | . . . 4 |
99 | 59, 91, 98 | 3eqtr4rd 2667 | . . 3 |
100 | 34, 99 | eqtr4d 2659 | . 2 |
101 | 10, 28, 100 | pm2.61da2ne 2882 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 crab 2916 cun 3572 cin 3573 wss 3574 c0 3915 class class class wbr 4653 cfv 5888 (class class class)co 6650 cfn 7955 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 cneg 10267 cn 11020 cn0 11292 cz 11377 cfz 12326 cexp 12860 chash 13117 cdvds 14983 cgcd 15216 cprime 15385 cmu 24821 |
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-rep 4771 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-int 4476 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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 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-q 11789 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-mu 24827 |
This theorem is referenced by: (None) |
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