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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8971 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | znnsub 9333 |
. . 3
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4 | zre 9286 |
. . . 4
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5 | zre 9286 |
. . . 4
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6 | 1re 7985 |
. . . . 5
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7 | leaddsub2 8425 |
. . . . 5
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8 | 6, 7 | mp3an2 1336 |
. . . 4
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9 | 4, 5, 8 | syl2an 289 |
. . 3
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10 | 2, 3, 9 | 3imtr4d 203 |
. 2
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11 | 4 | adantr 276 |
. . . 4
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12 | 11 | ltp1d 8916 |
. . 3
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13 | peano2re 8122 |
. . . . 5
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14 | 11, 13 | syl 14 |
. . . 4
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15 | 5 | adantl 277 |
. . . 4
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16 | ltletr 8076 |
. . . 4
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17 | 11, 14, 15, 16 | syl3anc 1249 |
. . 3
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18 | 12, 17 | mpand 429 |
. 2
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19 | 10, 18 | impbid 129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 |
This theorem is referenced by: zleltp1 9337 zlem1lt 9338 zgt0ge1 9340 nnltp1le 9342 nn0ltp1le 9344 btwnnz 9376 uzind2 9394 fzind 9397 btwnapz 9412 eluzp1l 9581 eluz2b1 9630 zltaddlt1le 10036 fzsplit2 10079 m1modge3gt1 10401 seq3f1olemqsumkj 10528 seq3f1olemqsumk 10529 bcval5 10774 seq3coll 10853 cvgratnnlemseq 11565 nn0o1gt2 11941 divalglemnqt 11956 zsupcllemstep 11977 infssuzex 11981 suprzubdc 11984 isprm3 12149 dvdsnprmd 12156 prmgt1 12163 oddprmge3 12166 znege1 12209 hashdvds 12252 lgsdilem2 14890 |
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