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Mirrors > Home > ILE Home > Th. List > letrd | GIF version |
Description: Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
letrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
letrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
letrd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | letrd.5 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
3 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | letrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | letr 7961 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1220 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 1, 2, 7 | mp2and 430 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2128 class class class wbr 3966 ℝcr 7732 ≤ cle 7914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltwlin 7846 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-cnv 4595 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 |
This theorem is referenced by: eluzuzle 9448 infregelbex 9510 fzdisj 9955 difelfzle 10037 flqwordi 10191 btwnzge0 10203 flqleceil 10220 modqltm1p1mod 10279 seq3split 10382 iseqf1olemqcl 10389 iseqf1olemnab 10391 iseqf1olemab 10392 seq3f1olemqsumkj 10401 seq3f1olemqsumk 10402 seq3f1olemqsum 10403 bernneq 10542 bernneq3 10544 nn0opthlem2d 10599 faclbnd 10619 facubnd 10623 seq3coll 10717 resqrexlemover 10914 resqrexlemdecn 10916 resqrexlemcalc3 10920 absle 10993 releabs 11000 maxleastb 11118 climsqz 11236 climsqz2 11237 fsum3cvg3 11297 expcnvap0 11403 geolim2 11413 cvgratnnlemabsle 11428 cvgratnnlemfm 11430 cvgratnnlemrate 11431 cvgratz 11433 mertenslem2 11437 eftlub 11591 cos12dec 11668 divalglemnqt 11815 infssuzex 11840 ncoprmgcdne1b 11970 prmdc 12011 eulerthlemrprm 12108 eulerthlema 12109 ennnfoneleminc 12182 ennnfonelemkh 12183 nninfdclemlt 12224 strleund 12320 suplociccex 13045 ivthinclemlopn 13056 ivthinclemuopn 13058 dveflem 13129 cosordlem 13212 rpabscxpbnd 13301 |
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