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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 5749 | . . . . 5 | |
3 | 2 | eqeq1d 2126 | . . . 4 |
4 | hashgcdlem.a | . . . 4 ..^ | |
5 | 3, 4 | elrab2 2816 | . . 3 ..^ |
6 | elfzonn0 9918 | . . . . . . 7 ..^ | |
7 | 6 | ad2antrl 481 | . . . . . 6 ..^ |
8 | nnnn0 8942 | . . . . . . . 8 | |
9 | 8 | 3ad2ant2 988 | . . . . . . 7 |
10 | 9 | adantr 274 | . . . . . 6 ..^ |
11 | 7, 10 | nn0mulcld 8993 | . . . . 5 ..^ |
12 | simpl1 969 | . . . . 5 ..^ | |
13 | elfzolt2 9888 | . . . . . . 7 ..^ | |
14 | 13 | ad2antrl 481 | . . . . . 6 ..^ |
15 | elfzoelz 9879 | . . . . . . . . 9 ..^ | |
16 | 15 | ad2antrl 481 | . . . . . . . 8 ..^ |
17 | 16 | zred 9131 | . . . . . . 7 ..^ |
18 | nnre 8691 | . . . . . . . . 9 | |
19 | 18 | 3ad2ant1 987 | . . . . . . . 8 |
20 | 19 | adantr 274 | . . . . . . 7 ..^ |
21 | nnre 8691 | . . . . . . . . . 10 | |
22 | nngt0 8709 | . . . . . . . . . 10 | |
23 | 21, 22 | jca 304 | . . . . . . . . 9 |
24 | 23 | 3ad2ant2 988 | . . . . . . . 8 |
25 | 24 | adantr 274 | . . . . . . 7 ..^ |
26 | ltmuldiv 8596 | . . . . . . 7 | |
27 | 17, 20, 25, 26 | syl3anc 1201 | . . . . . 6 ..^ |
28 | 14, 27 | mpbird 166 | . . . . 5 ..^ |
29 | elfzo0 9914 | . . . . 5 ..^ | |
30 | 11, 12, 28, 29 | syl3anbrc 1150 | . . . 4 ..^ ..^ |
31 | nncn 8692 | . . . . . . . . . 10 | |
32 | 31 | 3ad2ant1 987 | . . . . . . . . 9 |
33 | nncn 8692 | . . . . . . . . . 10 | |
34 | 33 | 3ad2ant2 988 | . . . . . . . . 9 |
35 | nnap0 8713 | . . . . . . . . . 10 # | |
36 | 35 | 3ad2ant2 988 | . . . . . . . . 9 # |
37 | 32, 34, 36 | divcanap1d 8518 | . . . . . . . 8 |
38 | 37 | adantr 274 | . . . . . . 7 ..^ |
39 | 38 | eqcomd 2123 | . . . . . 6 ..^ |
40 | 39 | oveq2d 5758 | . . . . 5 ..^ |
41 | nndivdvds 11411 | . . . . . . . . 9 | |
42 | 41 | biimp3a 1308 | . . . . . . . 8 |
43 | 42 | nnzd 9130 | . . . . . . 7 |
44 | 43 | adantr 274 | . . . . . 6 ..^ |
45 | mulgcdr 11618 | . . . . . 6 | |
46 | 16, 44, 10, 45 | syl3anc 1201 | . . . . 5 ..^ |
47 | simprr 506 | . . . . . . 7 ..^ | |
48 | 47 | oveq1d 5757 | . . . . . 6 ..^ |
49 | 34 | mulid2d 7752 | . . . . . . 7 |
50 | 49 | adantr 274 | . . . . . 6 ..^ |
51 | 48, 50 | eqtrd 2150 | . . . . 5 ..^ |
52 | 40, 46, 51 | 3eqtrd 2154 | . . . 4 ..^ |
53 | oveq1 5749 | . . . . . 6 | |
54 | 53 | eqeq1d 2126 | . . . . 5 |
55 | hashgcdlem.b | . . . . 5 ..^ | |
56 | 54, 55 | elrab2 2816 | . . . 4 ..^ |
57 | 30, 52, 56 | sylanbrc 413 | . . 3 ..^ |
58 | 5, 57 | sylan2b 285 | . 2 |
59 | oveq1 5749 | . . . . 5 | |
60 | 59 | eqeq1d 2126 | . . . 4 |
61 | 60, 55 | elrab2 2816 | . . 3 ..^ |
62 | simprr 506 | . . . . . . . 8 ..^ | |
63 | elfzoelz 9879 | . . . . . . . . . . 11 ..^ | |
64 | 63 | ad2antrl 481 | . . . . . . . . . 10 ..^ |
65 | simpl1 969 | . . . . . . . . . . 11 ..^ | |
66 | 65 | nnzd 9130 | . . . . . . . . . 10 ..^ |
67 | gcddvds 11564 | . . . . . . . . . 10 | |
68 | 64, 66, 67 | syl2anc 408 | . . . . . . . . 9 ..^ |
69 | 68 | simpld 111 | . . . . . . . 8 ..^ |
70 | 62, 69 | eqbrtrrd 3922 | . . . . . . 7 ..^ |
71 | nnz 9031 | . . . . . . . . . 10 | |
72 | 71 | 3ad2ant2 988 | . . . . . . . . 9 |
73 | 72 | adantr 274 | . . . . . . . 8 ..^ |
74 | nnne0 8712 | . . . . . . . . . 10 | |
75 | 74 | 3ad2ant2 988 | . . . . . . . . 9 |
76 | 75 | adantr 274 | . . . . . . . 8 ..^ |
77 | dvdsval2 11408 | . . . . . . . 8 | |
78 | 73, 76, 64, 77 | syl3anc 1201 | . . . . . . 7 ..^ |
79 | 70, 78 | mpbid 146 | . . . . . 6 ..^ |
80 | elfzofz 9894 | . . . . . . . . 9 ..^ | |
81 | 80 | ad2antrl 481 | . . . . . . . 8 ..^ |
82 | elfznn0 9849 | . . . . . . . 8 | |
83 | nn0re 8944 | . . . . . . . . 9 | |
84 | nn0ge0 8960 | . . . . . . . . 9 | |
85 | 83, 84 | jca 304 | . . . . . . . 8 |
86 | 81, 82, 85 | 3syl 17 | . . . . . . 7 ..^ |
87 | 24 | adantr 274 | . . . . . . 7 ..^ |
88 | divge0 8595 | . . . . . . 7 | |
89 | 86, 87, 88 | syl2anc 408 | . . . . . 6 ..^ |
90 | elnn0z 9025 | . . . . . 6 | |
91 | 79, 89, 90 | sylanbrc 413 | . . . . 5 ..^ |
92 | 42 | adantr 274 | . . . . 5 ..^ |
93 | elfzolt2 9888 | . . . . . . 7 ..^ | |
94 | 93 | ad2antrl 481 | . . . . . 6 ..^ |
95 | 64 | zred 9131 | . . . . . . 7 ..^ |
96 | 19 | adantr 274 | . . . . . . 7 ..^ |
97 | ltdiv1 8590 | . . . . . . 7 | |
98 | 95, 96, 87, 97 | syl3anc 1201 | . . . . . 6 ..^ |
99 | 94, 98 | mpbid 146 | . . . . 5 ..^ |
100 | elfzo0 9914 | . . . . 5 ..^ | |
101 | 91, 92, 99, 100 | syl3anbrc 1150 | . . . 4 ..^ ..^ |
102 | 62 | oveq1d 5757 | . . . . 5 ..^ |
103 | simpl2 970 | . . . . . 6 ..^ | |
104 | simpl3 971 | . . . . . 6 ..^ | |
105 | gcddiv 11619 | . . . . . 6 | |
106 | 64, 66, 103, 70, 104, 105 | syl32anc 1209 | . . . . 5 ..^ |
107 | 34, 36 | dividapd 8513 | . . . . . 6 |
108 | 107 | adantr 274 | . . . . 5 ..^ |
109 | 102, 106, 108 | 3eqtr3d 2158 | . . . 4 ..^ |
110 | oveq1 5749 | . . . . . 6 | |
111 | 110 | eqeq1d 2126 | . . . . 5 |
112 | 111, 4 | elrab2 2816 | . . . 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 9132 | . . . . . . 7 ..^ ..^ |
120 | 34 | adantr 274 | . . . . . . 7 ..^ ..^ |
121 | 36 | adantr 274 | . . . . . . 7 ..^ ..^ # |
122 | 119, 120, 121 | divcanap1d 8518 | . . . . . 6 ..^ ..^ |
123 | 122 | eqcomd 2123 | . . . . 5 ..^ ..^ |
124 | oveq1 5749 | . . . . . 6 | |
125 | 124 | eqeq2d 2129 | . . . . 5 |
126 | 123, 125 | syl5ibrcom 156 | . . . 4 ..^ ..^ |
127 | 15 | ad2antrl 481 | . . . . . . . 8 ..^ ..^ |
128 | 127 | zcnd 9132 | . . . . . . 7 ..^ ..^ |
129 | 128, 120, 121 | divcanap4d 8523 | . . . . . 6 ..^ ..^ |
130 | 129 | eqcomd 2123 | . . . . 5 ..^ ..^ |
131 | oveq1 5749 | . . . . . 6 | |
132 | 131 | eqeq2d 2129 | . . . . 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 5943 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wne 2285 crab 2397 class class class wbr 3899 cmpt 3959 wf1o 5092 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 cmul 7593 clt 7768 cle 7769 # cap 8310 cdiv 8399 cn 8684 cn0 8935 cz 9012 cfz 9745 ..^cfzo 9874 cdvds 11405 cgcd 11547 |
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-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 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-3 8744 df-4 8745 df-n0 8936 df-z 9013 df-uz 9283 df-q 9368 df-rp 9398 df-fz 9746 df-fzo 9875 df-fl 9998 df-mod 10051 df-seqfrec 10174 df-exp 10248 df-cj 10569 df-re 10570 df-im 10571 df-rsqrt 10725 df-abs 10726 df-dvds 11406 df-gcd 11548 |
This theorem is referenced by: hashgcdeq 11815 |
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