Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lelttr | Structured version Visualization version GIF version |
Description: Transitive law. (Contributed by NM, 23-May-1999.) |
Ref | Expression |
---|---|
lelttr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leloe 11061 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
2 | 1 | 3adant3 1131 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
3 | lttr 11051 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
4 | 3 | expd 416 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
5 | breq1 5077 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) | |
6 | 5 | biimprd 247 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)) |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
8 | 4, 7 | jaod 856 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
9 | 2, 8 | sylbid 239 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
10 | 9 | impd 411 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ℝcr 10870 < clt 11009 ≤ cle 11010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 |
This theorem is referenced by: leltletr 11066 letr 11069 lelttri 11102 lelttrd 11133 letrp1 11819 ltmul12a 11831 ledivp1 11877 supmul1 11944 bndndx 12232 uzind 12412 fnn0ind 12419 rpnnen1lem5 12721 xrinfmsslem 13042 elfzo0z 13429 nn0p1elfzo 13430 fzofzim 13434 elfzodifsumelfzo 13453 flge 13525 flflp1 13527 flltdivnn0lt 13553 modfzo0difsn 13663 fsequb 13695 expnlbnd2 13949 ccat2s1fvw 14349 ccat2s1fvwOLD 14350 swrdswrd 14418 pfxccatin12lem3 14445 repswswrd 14497 caubnd2 15069 caubnd 15070 mulcn2 15305 cn1lem 15307 rlimo1 15326 o1rlimmul 15328 climsqz 15350 climsqz2 15351 rlimsqzlem 15360 climsup 15381 caucvgrlem2 15386 iseralt 15396 cvgcmp 15528 cvgcmpce 15530 ruclem3 15942 ruclem12 15950 ltoddhalfle 16070 algcvgblem 16282 ncoprmlnprm 16432 pclem 16539 infpn2 16614 gsummoncoe1 21475 mp2pm2mplem4 21958 metss2lem 23667 ngptgp 23792 nghmcn 23909 iocopnst 24103 ovollb2lem 24652 ovolicc2lem4 24684 volcn 24770 ismbf3d 24818 dvcnvrelem1 25181 dvfsumrlim 25195 ulmcn 25558 mtest 25563 logdivlti 25775 isosctrlem1 25968 ftalem2 26223 chtub 26360 bposlem6 26437 gausslemma2dlem2 26515 chtppilim 26623 dchrisumlem3 26639 pntlem3 26757 clwlkclwwlklem2a 28362 vacn 29056 nmcvcn 29057 blocni 29167 chscllem2 30000 lnconi 30395 staddi 30608 stadd3i 30610 ltflcei 35765 poimirlem29 35806 geomcau 35917 heibor1lem 35967 bfplem2 35981 rrncmslem 35990 climinf 43147 zm1nn 44794 iccpartigtl 44875 tgoldbach 45269 ply1mulgsumlem2 45728 difmodm1lt 45868 |
Copyright terms: Public domain | W3C validator |