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Mirrors > Home > ILE Home > Th. List > moddvds | Unicode version |
Description: Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
moddvds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnq 9543 | . . . . . 6 | |
2 | 1 | adantr 274 | . . . . 5 |
3 | nngt0 8859 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | q0mod 10258 | . . . . 5 | |
6 | 2, 4, 5 | syl2anc 409 | . . . 4 |
7 | 6 | eqeq2d 2169 | . . 3 |
8 | zq 9536 | . . . . . . . . 9 | |
9 | 8 | ad2antrl 482 | . . . . . . . 8 |
10 | 9 | adantr 274 | . . . . . . 7 |
11 | zq 9536 | . . . . . . . . 9 | |
12 | 11 | ad2antll 483 | . . . . . . . 8 |
13 | 12 | adantr 274 | . . . . . . 7 |
14 | qnegcl 9546 | . . . . . . . 8 | |
15 | 13, 14 | syl 14 | . . . . . . 7 |
16 | 2 | adantr 274 | . . . . . . 7 |
17 | 4 | adantr 274 | . . . . . . 7 |
18 | simpr 109 | . . . . . . 7 | |
19 | 10, 13, 15, 16, 17, 18 | modqadd1 10264 | . . . . . 6 |
20 | 19 | ex 114 | . . . . 5 |
21 | simprl 521 | . . . . . . . . 9 | |
22 | 21 | zcnd 9288 | . . . . . . . 8 |
23 | simprr 522 | . . . . . . . . 9 | |
24 | 23 | zcnd 9288 | . . . . . . . 8 |
25 | 22, 24 | negsubd 8193 | . . . . . . 7 |
26 | 25 | oveq1d 5840 | . . . . . 6 |
27 | 24 | negidd 8177 | . . . . . . 7 |
28 | 27 | oveq1d 5840 | . . . . . 6 |
29 | 26, 28 | eqeq12d 2172 | . . . . 5 |
30 | 20, 29 | sylibd 148 | . . . 4 |
31 | 9 | adantr 274 | . . . . . . . 8 |
32 | 12 | adantr 274 | . . . . . . . 8 |
33 | qsubcl 9548 | . . . . . . . 8 | |
34 | 31, 32, 33 | syl2anc 409 | . . . . . . 7 |
35 | 0z 9179 | . . . . . . . 8 | |
36 | zq 9536 | . . . . . . . 8 | |
37 | 35, 36 | mp1i 10 | . . . . . . 7 |
38 | 2 | adantr 274 | . . . . . . 7 |
39 | 4 | adantr 274 | . . . . . . 7 |
40 | simpr 109 | . . . . . . 7 | |
41 | 34, 37, 32, 38, 39, 40 | modqadd1 10264 | . . . . . 6 |
42 | 41 | ex 114 | . . . . 5 |
43 | 22, 24 | npcand 8191 | . . . . . . 7 |
44 | 43 | oveq1d 5840 | . . . . . 6 |
45 | 24 | addid2d 8026 | . . . . . . 7 |
46 | 45 | oveq1d 5840 | . . . . . 6 |
47 | 44, 46 | eqeq12d 2172 | . . . . 5 |
48 | 42, 47 | sylibd 148 | . . . 4 |
49 | 30, 48 | impbid 128 | . . 3 |
50 | zsubcl 9209 | . . . 4 | |
51 | dvdsval3 11691 | . . . 4 | |
52 | 50, 51 | sylan2 284 | . . 3 |
53 | 7, 49, 52 | 3bitr4d 219 | . 2 |
54 | 53 | 3impb 1181 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 class class class wbr 3966 (class class class)co 5825 cc0 7733 caddc 7736 clt 7913 cmin 8047 cneg 8048 cn 8834 cz 9168 cq 9529 cmo 10225 cdvds 11687 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-po 4257 df-iso 4258 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-n0 9092 df-z 9169 df-q 9530 df-rp 9562 df-fl 10173 df-mod 10226 df-dvds 11688 |
This theorem is referenced by: modm1div 11700 summodnegmod 11722 modmulconst 11723 addmodlteqALT 11755 dvdsmod 11758 congr 11981 cncongr1 11984 cncongr2 11985 crth 12103 eulerthlemh 12110 eulerthlemth 12111 prmdiv 12114 prmdiveq 12115 odzcllem 12121 odzdvds 12124 odzphi 12125 |
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