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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 8043 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1238 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 1, 2, 7 | mp2and 433 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4005 ℝcr 7813 ≤ cle 7996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-pre-ltwlin 7927 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 |
This theorem is referenced by: nn0negleid 9324 eluzuzle 9539 infregelbex 9601 fzdisj 10055 difelfzle 10137 flqwordi 10291 btwnzge0 10303 flqleceil 10320 modqltm1p1mod 10379 seq3split 10482 iseqf1olemqcl 10489 iseqf1olemnab 10491 iseqf1olemab 10492 seq3f1olemqsumkj 10501 seq3f1olemqsumk 10502 seq3f1olemqsum 10503 bernneq 10644 bernneq3 10646 nn0opthlem2d 10704 faclbnd 10724 facubnd 10728 seq3coll 10825 resqrexlemover 11022 resqrexlemdecn 11024 resqrexlemcalc3 11028 absle 11101 releabs 11108 maxleastb 11226 climsqz 11346 climsqz2 11347 fsum3cvg3 11407 expcnvap0 11513 geolim2 11523 cvgratnnlemabsle 11538 cvgratnnlemfm 11540 cvgratnnlemrate 11541 cvgratz 11543 mertenslem2 11547 eftlub 11701 cos12dec 11778 divalglemnqt 11928 infssuzex 11953 suprzubdc 11956 ncoprmgcdne1b 12092 prmdc 12133 isprm5lem 12144 eulerthlemrprm 12232 eulerthlema 12233 pcmpt2 12345 pcfac 12351 ennnfoneleminc 12415 ennnfonelemkh 12416 nninfdclemlt 12455 strleund 12565 strext 12567 suplociccex 14243 ivthinclemlopn 14254 ivthinclemuopn 14256 dveflem 14327 cosordlem 14410 rpabscxpbnd 14499 lgsdirprm 14575 2sqlem8 14610 |
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