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Mirrors > Home > MPE Home > Th. List > ltletr | Structured version Visualization version Unicode version |
Description: Transitive law. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
ltletr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leloe 10124 | . . . . 5 | |
2 | 1 | 3adant1 1079 | . . . 4 |
3 | lttr 10114 | . . . . . 6 | |
4 | 3 | expcomd 454 | . . . . 5 |
5 | breq2 4657 | . . . . . . 7 | |
6 | 5 | biimpd 219 | . . . . . 6 |
7 | 6 | a1i 11 | . . . . 5 |
8 | 4, 7 | jaod 395 | . . . 4 |
9 | 2, 8 | sylbid 230 | . . 3 |
10 | 9 | com23 86 | . 2 |
11 | 10 | impd 447 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 cr 9935 clt 10074 cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 |
This theorem is referenced by: ltleletr 10130 ltletri 10165 ltletrd 10197 ltleadd 10511 lediv12a 10916 nngt0 11049 nnrecgt0 11058 elnnnn0c 11338 elnnz1 11403 zltp1le 11427 uz3m2nn 11731 zbtwnre 11786 ledivge1le 11901 addlelt 11942 qbtwnre 12030 xlemul1a 12118 xrsupsslem 12137 zltaddlt1le 12324 elfzodifsumelfzo 12533 ssfzo12bi 12563 elfznelfzo 12573 ceile 12648 swrdswrd 13460 swrdccatin1 13483 repswswrd 13531 sqrlem4 13986 resqrex 13991 caubnd 14098 rlim2lt 14228 cos01gt0 14921 znnenlem 14940 ruclem12 14970 oddge22np1 15073 sadcaddlem 15179 nn0seqcvgd 15283 coprm 15423 prmgaplem7 15761 prmlem1 15814 prmlem2 15827 icoopnst 22738 ovollb2lem 23256 dvcnvrelem1 23780 aaliou 24093 tanord 24284 logdivlti 24366 logdivlt 24367 ftalem2 24800 gausslemma2dlem1a 25090 pntlem3 25298 crctcshwlkn0lem3 26704 numclwlk1lem2f1 27227 nn0prpwlem 32317 isbasisrelowllem1 33203 isbasisrelowllem2 33204 ltflcei 33397 tan2h 33401 poimirlem29 33438 poimirlem32 33441 stoweidlem26 40243 stoweid 40280 2leaddle2 41312 pfx2 41412 gbegt5 41649 gbowgt5 41650 sgoldbeven3prm 41671 nnsum4primesodd 41684 nnsum4primesoddALTV 41685 evengpoap3 41687 bgoldbnnsum3prm 41692 cznnring 41956 nn0sumltlt 42128 rege1logbrege0 42352 rege1logbzge0 42353 fllog2 42362 dignn0ldlem 42396 |
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