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Mirrors > Home > ILE Home > Th. List > zltp1le | Unicode version |
Description: Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zltp1le |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8743 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | znnsub 9105 | . . 3 | |
4 | zre 9058 | . . . 4 | |
5 | zre 9058 | . . . 4 | |
6 | 1re 7765 | . . . . 5 | |
7 | leaddsub2 8201 | . . . . 5 | |
8 | 6, 7 | mp3an2 1303 | . . . 4 |
9 | 4, 5, 8 | syl2an 287 | . . 3 |
10 | 2, 3, 9 | 3imtr4d 202 | . 2 |
11 | 4 | adantr 274 | . . . 4 |
12 | 11 | ltp1d 8688 | . . 3 |
13 | peano2re 7898 | . . . . 5 | |
14 | 11, 13 | syl 14 | . . . 4 |
15 | 5 | adantl 275 | . . . 4 |
16 | ltletr 7853 | . . . 4 | |
17 | 11, 14, 15, 16 | syl3anc 1216 | . . 3 |
18 | 12, 17 | mpand 425 | . 2 |
19 | 10, 18 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 clt 7800 cle 7801 cmin 7933 cn 8720 cz 9054 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 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-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 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-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 |
This theorem is referenced by: zleltp1 9109 zlem1lt 9110 zgt0ge1 9112 nnltp1le 9114 nn0ltp1le 9116 btwnnz 9145 uzind2 9163 fzind 9166 btwnapz 9181 eluzp1l 9350 eluz2b1 9395 zltaddlt1le 9789 fzsplit2 9830 m1modge3gt1 10144 seq3f1olemqsumkj 10271 seq3f1olemqsumk 10272 bcval5 10509 seq3coll 10585 cvgratnnlemseq 11295 nn0o1gt2 11602 divalglemnqt 11617 zsupcllemstep 11638 infssuzex 11642 isprm3 11799 dvdsnprmd 11806 prmgt1 11812 oddprmge3 11815 znege1 11856 hashdvds 11897 |
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