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Mirrors > Home > MPE Home > Th. List > letr | Structured version Visualization version GIF version |
Description: Transitive law. (Contributed by NM, 12-Nov-1999.) |
Ref | Expression |
---|---|
letr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leloe 10716 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) | |
2 | 1 | 3adant1 1122 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) |
3 | 2 | adantr 481 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) |
4 | lelttr 10720 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
5 | ltle 10718 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) | |
6 | 5 | 3adant2 1123 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) |
7 | 4, 6 | syld 47 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 7 | expdimp 453 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐵 < 𝐶 → 𝐴 ≤ 𝐶)) |
9 | breq2 5062 | . . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ 𝐶)) | |
10 | 9 | biimpcd 250 | . . . . 5 ⊢ (𝐴 ≤ 𝐵 → (𝐵 = 𝐶 → 𝐴 ≤ 𝐶)) |
11 | 10 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐵 = 𝐶 → 𝐴 ≤ 𝐶)) |
12 | 8, 11 | jaod 853 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → ((𝐵 < 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ≤ 𝐶)) |
13 | 3, 12 | sylbid 241 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → (𝐵 ≤ 𝐶 → 𝐴 ≤ 𝐶)) |
14 | 13 | expimpd 454 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ℝcr 10525 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: letri 10758 letrd 10786 le2add 11111 le2sub 11128 p1le 11474 lemul12b 11486 lemul12a 11487 zletr 12015 peano2uz2 12059 ledivge1le 12450 lemaxle 12578 elfz1b 12966 elfz0fzfz0 13002 fz0fzelfz0 13003 fz0fzdiffz0 13006 elfzmlbp 13008 difelfznle 13011 elincfzoext 13085 ssfzoulel 13121 ssfzo12bi 13122 flge 13165 flflp1 13167 fldiv4p1lem1div2 13195 fldiv4lem1div2uz2 13196 monoord 13390 le2sq2 13490 leexp2r 13528 expubnd 13531 facwordi 13639 faclbnd3 13642 facavg 13651 fi1uzind 13845 swrdswrdlem 14056 swrdccat 14087 sqrlem1 14592 sqrlem6 14597 sqrlem7 14598 leabs 14649 limsupbnd2 14830 rlim3 14845 lo1bdd2 14871 lo1bddrp 14872 o1lo1 14884 lo1mul 14974 lo1le 14998 isercolllem2 15012 iseraltlem2 15029 fsumabs 15146 cvgrat 15229 ruclem9 15581 algcvga 15913 prmdvdsfz 16039 prmfac1 16053 eulerthlem2 16109 modprm0 16132 prmreclem1 16242 prmreclem4 16245 4sqlem11 16281 vdwnnlem3 16323 gsumbagdiaglem 20085 zntoslem 20633 cnllycmp 23489 evth 23492 ovoliunlem2 24033 ovolicc2lem3 24049 itg2monolem1 24280 coeaddlem 24768 coemullem 24769 aalioulem5 24854 aalioulem6 24855 sincosq1lem 25012 emcllem6 25506 ftalem3 25580 fsumvma2 25718 chpchtsum 25723 bcmono 25781 bposlem5 25792 gausslemma2dlem1a 25869 lgsquadlem1 25884 dchrisum0lem1 26020 pntrsumbnd2 26071 pntleml 26115 brbtwn2 26619 axlowdimlem17 26672 axlowdim 26675 crctcshwlkn0lem3 27518 crctcshwlkn0lem5 27520 wwlksubclwwlk 27765 eupth2lems 27945 nmoub3i 28478 ubthlem1 28575 ubthlem2 28576 nmopub2tALT 29614 nmfnleub2 29631 lnconi 29738 leoptr 29842 pjnmopi 29853 cdj3lem2b 30142 eulerpartlemb 31526 isbasisrelowllem1 34519 isbasisrelowllem2 34520 ltflcei 34762 itg2addnclem2 34826 itg2addnclem3 34827 itg2addnc 34828 bddiblnc 34844 dvasin 34860 incsequz 34906 mettrifi 34915 equivbnd 34951 bfplem1 34983 jm2.17b 39438 fmul01lt1lem2 41746 eluzge0nn0 43393 elfz2z 43396 iccpartiltu 43429 iccpartgt 43434 lighneallem2 43618 |
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