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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 |
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ltadd2d.2 |
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ltadd2d.3 |
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ltletrd.4 |
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ltletrd.5 |
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Ref | Expression |
---|---|
ltletrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 |
. 2
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2 | ltletrd.5 |
. 2
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3 | ltadd2d.1 |
. . 3
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4 | ltadd2d.2 |
. . 3
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5 | ltadd2d.3 |
. . 3
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6 | ltletr 8045 |
. . 3
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7 | 3, 4, 5, 6 | syl3anc 1238 |
. 2
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8 | 1, 2, 7 | mp2and 433 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltwlin 7923 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 |
This theorem is referenced by: lelttrdi 8381 lediv12a 8849 btwnapz 9381 rpgecl 9680 fznatpl1 10073 elfz1b 10087 exbtwnzlemstep 10245 ceiqle 10310 modqabs 10354 mulp1mod1 10362 seq3f1olemqsumk 10496 expgt1 10555 leexp2a 10570 bernneq3 10639 expnbnd 10640 nn0opthlem2d 10696 cvg1nlemres 10989 resqrexlemlo 11017 resqrexlemnmsq 11021 resqrexlemga 11027 abssubap0 11094 icodiamlt 11184 rpmaxcl 11227 reccn2ap 11316 divcnv 11500 cvgratnnlembern 11526 cvgratnnlemabsle 11530 fprodntrivap 11587 efcllemp 11661 sin01bnd 11760 cos01bnd 11761 sin01gt0 11764 cos12dec 11770 eirraplem 11779 dvdslelemd 11843 dvdsbnd 11951 isprm5 12136 1arith 12359 znnen 12393 nninfdclemp1 12445 cnopnap 13987 dedekindeulemlu 13992 suplociccreex 13995 dedekindicclemlu 14001 dedekindicc 14004 ivthinclemlopn 14007 limcimolemlt 14026 limccnp2lem 14038 coseq00topi 14149 cosordlem 14163 logdivlti 14195 |
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