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Mirrors > Home > ILE Home > Th. List > hashgcdlem | Unicode version |
Description: A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
hashgcdlem.a | ..^ |
hashgcdlem.b | ..^ |
hashgcdlem.f |
Ref | Expression |
---|---|
hashgcdlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashgcdlem.f | . 2 | |
2 | oveq1 5781 | . . . . 5 | |
3 | 2 | eqeq1d 2148 | . . . 4 |
4 | hashgcdlem.a | . . . 4 ..^ | |
5 | 3, 4 | elrab2 2843 | . . 3 ..^ |
6 | elfzonn0 9963 | . . . . . . 7 ..^ | |
7 | 6 | ad2antrl 481 | . . . . . 6 ..^ |
8 | nnnn0 8984 | . . . . . . . 8 | |
9 | 8 | 3ad2ant2 1003 | . . . . . . 7 |
10 | 9 | adantr 274 | . . . . . 6 ..^ |
11 | 7, 10 | nn0mulcld 9035 | . . . . 5 ..^ |
12 | simpl1 984 | . . . . 5 ..^ | |
13 | elfzolt2 9933 | . . . . . . 7 ..^ | |
14 | 13 | ad2antrl 481 | . . . . . 6 ..^ |
15 | elfzoelz 9924 | . . . . . . . . 9 ..^ | |
16 | 15 | ad2antrl 481 | . . . . . . . 8 ..^ |
17 | 16 | zred 9173 | . . . . . . 7 ..^ |
18 | nnre 8727 | . . . . . . . . 9 | |
19 | 18 | 3ad2ant1 1002 | . . . . . . . 8 |
20 | 19 | adantr 274 | . . . . . . 7 ..^ |
21 | nnre 8727 | . . . . . . . . . 10 | |
22 | nngt0 8745 | . . . . . . . . . 10 | |
23 | 21, 22 | jca 304 | . . . . . . . . 9 |
24 | 23 | 3ad2ant2 1003 | . . . . . . . 8 |
25 | 24 | adantr 274 | . . . . . . 7 ..^ |
26 | ltmuldiv 8632 | . . . . . . 7 | |
27 | 17, 20, 25, 26 | syl3anc 1216 | . . . . . 6 ..^ |
28 | 14, 27 | mpbird 166 | . . . . 5 ..^ |
29 | elfzo0 9959 | . . . . 5 ..^ | |
30 | 11, 12, 28, 29 | syl3anbrc 1165 | . . . 4 ..^ ..^ |
31 | nncn 8728 | . . . . . . . . . 10 | |
32 | 31 | 3ad2ant1 1002 | . . . . . . . . 9 |
33 | nncn 8728 | . . . . . . . . . 10 | |
34 | 33 | 3ad2ant2 1003 | . . . . . . . . 9 |
35 | nnap0 8749 | . . . . . . . . . 10 # | |
36 | 35 | 3ad2ant2 1003 | . . . . . . . . 9 # |
37 | 32, 34, 36 | divcanap1d 8551 | . . . . . . . 8 |
38 | 37 | adantr 274 | . . . . . . 7 ..^ |
39 | 38 | eqcomd 2145 | . . . . . 6 ..^ |
40 | 39 | oveq2d 5790 | . . . . 5 ..^ |
41 | nndivdvds 11499 | . . . . . . . . 9 | |
42 | 41 | biimp3a 1323 | . . . . . . . 8 |
43 | 42 | nnzd 9172 | . . . . . . 7 |
44 | 43 | adantr 274 | . . . . . 6 ..^ |
45 | mulgcdr 11706 | . . . . . 6 | |
46 | 16, 44, 10, 45 | syl3anc 1216 | . . . . 5 ..^ |
47 | simprr 521 | . . . . . . 7 ..^ | |
48 | 47 | oveq1d 5789 | . . . . . 6 ..^ |
49 | 34 | mulid2d 7784 | . . . . . . 7 |
50 | 49 | adantr 274 | . . . . . 6 ..^ |
51 | 48, 50 | eqtrd 2172 | . . . . 5 ..^ |
52 | 40, 46, 51 | 3eqtrd 2176 | . . . 4 ..^ |
53 | oveq1 5781 | . . . . . 6 | |
54 | 53 | eqeq1d 2148 | . . . . 5 |
55 | hashgcdlem.b | . . . . 5 ..^ | |
56 | 54, 55 | elrab2 2843 | . . . 4 ..^ |
57 | 30, 52, 56 | sylanbrc 413 | . . 3 ..^ |
58 | 5, 57 | sylan2b 285 | . 2 |
59 | oveq1 5781 | . . . . 5 | |
60 | 59 | eqeq1d 2148 | . . . 4 |
61 | 60, 55 | elrab2 2843 | . . 3 ..^ |
62 | simprr 521 | . . . . . . . 8 ..^ | |
63 | elfzoelz 9924 | . . . . . . . . . . 11 ..^ | |
64 | 63 | ad2antrl 481 | . . . . . . . . . 10 ..^ |
65 | simpl1 984 | . . . . . . . . . . 11 ..^ | |
66 | 65 | nnzd 9172 | . . . . . . . . . 10 ..^ |
67 | gcddvds 11652 | . . . . . . . . . 10 | |
68 | 64, 66, 67 | syl2anc 408 | . . . . . . . . 9 ..^ |
69 | 68 | simpld 111 | . . . . . . . 8 ..^ |
70 | 62, 69 | eqbrtrrd 3952 | . . . . . . 7 ..^ |
71 | nnz 9073 | . . . . . . . . . 10 | |
72 | 71 | 3ad2ant2 1003 | . . . . . . . . 9 |
73 | 72 | adantr 274 | . . . . . . . 8 ..^ |
74 | nnne0 8748 | . . . . . . . . . 10 | |
75 | 74 | 3ad2ant2 1003 | . . . . . . . . 9 |
76 | 75 | adantr 274 | . . . . . . . 8 ..^ |
77 | dvdsval2 11496 | . . . . . . . 8 | |
78 | 73, 76, 64, 77 | syl3anc 1216 | . . . . . . 7 ..^ |
79 | 70, 78 | mpbid 146 | . . . . . 6 ..^ |
80 | elfzofz 9939 | . . . . . . . . 9 ..^ | |
81 | 80 | ad2antrl 481 | . . . . . . . 8 ..^ |
82 | elfznn0 9894 | . . . . . . . 8 | |
83 | nn0re 8986 | . . . . . . . . 9 | |
84 | nn0ge0 9002 | . . . . . . . . 9 | |
85 | 83, 84 | jca 304 | . . . . . . . 8 |
86 | 81, 82, 85 | 3syl 17 | . . . . . . 7 ..^ |
87 | 24 | adantr 274 | . . . . . . 7 ..^ |
88 | divge0 8631 | . . . . . . 7 | |
89 | 86, 87, 88 | syl2anc 408 | . . . . . 6 ..^ |
90 | elnn0z 9067 | . . . . . 6 | |
91 | 79, 89, 90 | sylanbrc 413 | . . . . 5 ..^ |
92 | 42 | adantr 274 | . . . . 5 ..^ |
93 | elfzolt2 9933 | . . . . . . 7 ..^ | |
94 | 93 | ad2antrl 481 | . . . . . 6 ..^ |
95 | 64 | zred 9173 | . . . . . . 7 ..^ |
96 | 19 | adantr 274 | . . . . . . 7 ..^ |
97 | ltdiv1 8626 | . . . . . . 7 | |
98 | 95, 96, 87, 97 | syl3anc 1216 | . . . . . 6 ..^ |
99 | 94, 98 | mpbid 146 | . . . . 5 ..^ |
100 | elfzo0 9959 | . . . . 5 ..^ | |
101 | 91, 92, 99, 100 | syl3anbrc 1165 | . . . 4 ..^ ..^ |
102 | 62 | oveq1d 5789 | . . . . 5 ..^ |
103 | simpl2 985 | . . . . . 6 ..^ | |
104 | simpl3 986 | . . . . . 6 ..^ | |
105 | gcddiv 11707 | . . . . . 6 | |
106 | 64, 66, 103, 70, 104, 105 | syl32anc 1224 | . . . . 5 ..^ |
107 | 34, 36 | dividapd 8546 | . . . . . 6 |
108 | 107 | adantr 274 | . . . . 5 ..^ |
109 | 102, 106, 108 | 3eqtr3d 2180 | . . . 4 ..^ |
110 | oveq1 5781 | . . . . . 6 | |
111 | 110 | eqeq1d 2148 | . . . . 5 |
112 | 111, 4 | elrab2 2843 | . . . 4 ..^ |
113 | 101, 109, 112 | sylanbrc 413 | . . 3 ..^ |
114 | 61, 113 | sylan2b 285 | . 2 |
115 | 5 | simplbi 272 | . . . 4 ..^ |
116 | 61 | simplbi 272 | . . . 4 ..^ |
117 | 115, 116 | anim12i 336 | . . 3 ..^ ..^ |
118 | 63 | ad2antll 482 | . . . . . . . 8 ..^ ..^ |
119 | 118 | zcnd 9174 | . . . . . . 7 ..^ ..^ |
120 | 34 | adantr 274 | . . . . . . 7 ..^ ..^ |
121 | 36 | adantr 274 | . . . . . . 7 ..^ ..^ # |
122 | 119, 120, 121 | divcanap1d 8551 | . . . . . 6 ..^ ..^ |
123 | 122 | eqcomd 2145 | . . . . 5 ..^ ..^ |
124 | oveq1 5781 | . . . . . 6 | |
125 | 124 | eqeq2d 2151 | . . . . 5 |
126 | 123, 125 | syl5ibrcom 156 | . . . 4 ..^ ..^ |
127 | 15 | ad2antrl 481 | . . . . . . . 8 ..^ ..^ |
128 | 127 | zcnd 9174 | . . . . . . 7 ..^ ..^ |
129 | 128, 120, 121 | divcanap4d 8556 | . . . . . 6 ..^ ..^ |
130 | 129 | eqcomd 2145 | . . . . 5 ..^ ..^ |
131 | oveq1 5781 | . . . . . 6 | |
132 | 131 | eqeq2d 2151 | . . . . 5 |
133 | 130, 132 | syl5ibrcom 156 | . . . 4 ..^ ..^ |
134 | 126, 133 | impbid 128 | . . 3 ..^ ..^ |
135 | 117, 134 | sylan2 284 | . 2 |
136 | 1, 58, 114, 135 | f1o2d 5975 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wne 2308 crab 2420 class class class wbr 3929 cmpt 3989 wf1o 5122 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 cmul 7625 clt 7800 cle 7801 # cap 8343 cdiv 8432 cn 8720 cn0 8977 cz 9054 cfz 9790 ..^cfzo 9919 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-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: hashgcdeq 11904 |
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