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| Mirrors > Home > MPE Home > Th. List > lttri | Structured version Visualization version GIF version | ||
| Description: 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
| Ref | Expression |
|---|---|
| lt.1 | ⊢ 𝐴 ∈ ℝ |
| lt.2 | ⊢ 𝐵 ∈ ℝ |
| lt.3 | ⊢ 𝐶 ∈ ℝ |
| Ref | Expression |
|---|---|
| lttri | ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lt.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
| 2 | lt.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
| 3 | lt.3 | . 2 ⊢ 𝐶 ∈ ℝ | |
| 4 | lttr 11337 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
| 5 | 1, 2, 3, 4 | mp3an 1463 | 1 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: 1lt3 12439 2lt4 12441 1lt4 12442 3lt5 12444 2lt5 12445 1lt5 12446 4lt6 12448 3lt6 12449 2lt6 12450 1lt6 12451 5lt7 12453 4lt7 12454 3lt7 12455 2lt7 12456 1lt7 12457 6lt8 12459 5lt8 12460 4lt8 12461 3lt8 12462 2lt8 12463 1lt8 12464 7lt9 12466 6lt9 12467 5lt9 12468 4lt9 12469 3lt9 12470 2lt9 12471 1lt9 12472 8lt10 12865 7lt10 12866 6lt10 12867 5lt10 12868 4lt10 12869 3lt10 12870 2lt10 12871 1lt10 12872 sincos2sgn 16230 epos 16243 ene1 16246 dvdslelem 16346 sralemOLD 21176 zlmlemOLD 21528 psgnodpmr 21608 tnglemOLD 24654 xrhmph 24978 vitalilem4 25646 pipos 26502 logi 26629 logneg 26630 asin1 26937 reasinsin 26939 atan1 26971 log2le1 26993 bposlem8 27335 bposlem9 27336 chebbnd1lem2 27514 chebbnd1lem3 27515 chebbnd1 27516 mulog2sumlem2 27579 pntibndlem1 27633 pntlemb 27641 pntlemk 27650 ttglemOLD 28886 cchhllemOLD 28902 axlowdimlem16 28972 dp2ltc 32869 sgnnbi 34548 sgnpbi 34549 signswch 34576 hgt750lem 34666 hgt750lem2 34667 cnndvlem1 36538 bj-minftyccb 37226 bj-pinftynminfty 37228 irrdiff 37327 asindmre 37710 fdc 37752 lttrii 42297 sn-0ne2 42436 fourierdlem94 46215 fourierdlem102 46223 fourierdlem103 46224 fourierdlem104 46225 fourierdlem112 46233 fourierdlem113 46234 fourierdlem114 46235 fouriersw 46246 etransclem23 46272 |
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