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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 7847 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1216 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 1, 2, 7 | mp2and 429 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 ≤ cle 7801 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltwlin 7733 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: eluzuzle 9334 fzdisj 9832 difelfzle 9911 flqwordi 10061 btwnzge0 10073 flqleceil 10090 modqltm1p1mod 10149 seq3split 10252 iseqf1olemqcl 10259 iseqf1olemnab 10261 iseqf1olemab 10262 seq3f1olemqsumkj 10271 seq3f1olemqsumk 10272 seq3f1olemqsum 10273 bernneq 10412 bernneq3 10414 nn0opthlem2d 10467 faclbnd 10487 facubnd 10491 seq3coll 10585 resqrexlemover 10782 resqrexlemdecn 10784 resqrexlemcalc3 10788 absle 10861 releabs 10868 maxleastb 10986 climsqz 11104 climsqz2 11105 fsum3cvg3 11165 expcnvap0 11271 geolim2 11281 cvgratnnlemabsle 11296 cvgratnnlemfm 11298 cvgratnnlemrate 11299 cvgratz 11301 mertenslem2 11305 eftlub 11396 cos12dec 11474 divalglemnqt 11617 infssuzex 11642 ncoprmgcdne1b 11770 ennnfoneleminc 11924 ennnfonelemkh 11925 strleund 12047 suplociccex 12772 ivthinclemlopn 12783 ivthinclemuopn 12785 dveflem 12855 cosordlem 12930 |
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