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Mirrors > Home > MPE Home > Th. List > zltlem1 | Structured version Visualization version Unicode version |
Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
Ref | Expression |
---|---|
zltlem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 11008 |
. . 3
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2 | zleltp1 11015 |
. . 3
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3 | 1, 2 | sylan2 481 |
. 2
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4 | zcn 10970 |
. . . . 5
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5 | ax-1cn 9622 |
. . . . 5
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6 | npcan 9909 |
. . . . 5
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7 | 4, 5, 6 | sylancl 673 |
. . . 4
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8 | 7 | adantl 472 |
. . 3
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9 | 8 | breq2d 4427 |
. 2
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10 | 3, 9 | bitr2d 262 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-om 6719 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-er 7388 df-en 7595 df-dom 7596 df-sdom 7597 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-nn 10637 df-n0 10898 df-z 10966 |
This theorem is referenced by: nn0ltlem1 11024 nn0lt2 11028 nnltlem1 11031 nnm1ge0 11032 zextlt 11038 uzm1 11217 elfzm11 11893 preduz 11941 predfz 11944 elfzo 11952 fzosplitprm1 12048 intfracq 12117 seqf1olem1 12283 seqcoll 12659 isercolllem1 13776 fzm1ndvds 14405 bitscmp 14460 nn0seqcvgd 14577 isprm3 14681 ncoprmlnprm 14725 prmdiveq 14782 4sqlem12 14948 degltlem1 23069 dgreq0 23267 wilthlem1 24041 lgseisenlem2 24326 lgsquadlem1 24330 2sqlem8 24348 clwlkisclwwlklem2a4 25560 clwlkisclwwlklem2a 25561 nbhashuvtx1 25691 frgrareggt1 25892 bcm1n 28419 smatrcl 28670 ballotlemimin 29386 ballotlemfrcn0 29410 ballotlemiminOLD 29424 ballotlemfrcn0OLD 29448 poimirlem2 31986 poimirlem24 32008 fmul01lt1lem2 37700 fourierdlem41 38048 fourierdlem42 38049 fourierdlem42OLD 38050 fourierdlem50 38057 fourierdlem64 38071 fourierdlem79 38086 etransclem44 38180 etransclem48OLD 38184 etransclem48 38185 nn0le2is012 40420 pw2m1lepw2m1 40590 fllog2 40651 |
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