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Mirrors > Home > ILE Home > Th. List > zdiv | Unicode version |
Description: Two ways to express " divides . (Contributed by NM, 3-Oct-2008.) |
Ref | Expression |
---|---|
zdiv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnap0 8868 | . . 3 # | |
2 | 1 | adantr 274 | . 2 # |
3 | nncn 8847 | . . 3 | |
4 | zcn 9178 | . . 3 | |
5 | zcn 9178 | . . . . . . . . . . 11 | |
6 | divcanap3 8576 | . . . . . . . . . . . . 13 # | |
7 | 6 | 3coml 1192 | . . . . . . . . . . . 12 # |
8 | 7 | 3expa 1185 | . . . . . . . . . . 11 # |
9 | 5, 8 | sylan2 284 | . . . . . . . . . 10 # |
10 | 9 | 3adantl2 1139 | . . . . . . . . 9 # |
11 | oveq1 5834 | . . . . . . . . 9 | |
12 | 10, 11 | sylan9req 2211 | . . . . . . . 8 # |
13 | simplr 520 | . . . . . . . 8 # | |
14 | 12, 13 | eqeltrrd 2235 | . . . . . . 7 # |
15 | 14 | exp31 362 | . . . . . 6 # |
16 | 15 | rexlimdv 2573 | . . . . 5 # |
17 | divcanap2 8558 | . . . . . . 7 # | |
18 | 17 | 3com12 1189 | . . . . . 6 # |
19 | oveq2 5835 | . . . . . . . . 9 | |
20 | 19 | eqeq1d 2166 | . . . . . . . 8 |
21 | 20 | rspcev 2816 | . . . . . . 7 |
22 | 21 | expcom 115 | . . . . . 6 |
23 | 18, 22 | syl 14 | . . . . 5 # |
24 | 16, 23 | impbid 128 | . . . 4 # |
25 | 24 | 3expia 1187 | . . 3 # |
26 | 3, 4, 25 | syl2an 287 | . 2 # |
27 | 2, 26 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wrex 2436 class class class wbr 3967 (class class class)co 5827 cc 7733 cc0 7735 cmul 7740 # cap 8461 cdiv 8550 cn 8839 cz 9173 |
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 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 |
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-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-id 4256 df-po 4259 df-iso 4260 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-z 9174 |
This theorem is referenced by: addmodlteq 10307 |
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