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Mirrors > Home > ILE Home > Th. List > ltletrd | Unicode version |
Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltadd2d.1 | |
ltadd2d.2 | |
ltadd2d.3 | |
ltletrd.4 | |
ltletrd.5 |
Ref | Expression |
---|---|
ltletrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 | |
2 | ltletrd.5 | . 2 | |
3 | ltadd2d.1 | . . 3 | |
4 | ltadd2d.2 | . . 3 | |
5 | ltadd2d.3 | . . 3 | |
6 | ltletr 7846 | . . 3 | |
7 | 3, 4, 5, 6 | syl3anc 1216 | . 2 |
8 | 1, 2, 7 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 class class class wbr 3924 cr 7612 clt 7793 cle 7794 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltwlin 7726 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: lelttrdi 8181 lediv12a 8645 btwnapz 9174 rpgecl 9463 fznatpl1 9849 elfz1b 9863 exbtwnzlemstep 10018 ceiqle 10079 modqabs 10123 mulp1mod1 10131 seq3f1olemqsumk 10265 expgt1 10324 leexp2a 10339 bernneq3 10407 expnbnd 10408 nn0opthlem2d 10460 cvg1nlemres 10750 resqrexlemlo 10778 resqrexlemnmsq 10782 resqrexlemga 10788 abssubap0 10855 icodiamlt 10945 rpmaxcl 10988 reccn2ap 11075 divcnv 11259 cvgratnnlembern 11285 cvgratnnlemabsle 11289 efcllemp 11353 sin01bnd 11453 cos01bnd 11454 sin01gt0 11457 cos12dec 11463 eirraplem 11472 dvdslelemd 11530 dvdsbnd 11634 znnen 11900 cnopnap 12752 dedekindeulemlu 12757 suplociccreex 12760 dedekindicclemlu 12766 dedekindicc 12769 ivthinclemlopn 12772 limcimolemlt 12791 limccnp2lem 12803 coseq00topi 12905 cosordlem 12919 |
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