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Mirrors > Home > ILE Home > Th. List > ltletrd | GIF version |
Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltadd2d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltadd2d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd2d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
ltletrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
ltletrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltletrd.5 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
3 | ltadd2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltadd2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | ltadd2d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | ltletr 7979 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1227 | . 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 2135 class class class wbr 3976 ℝcr 7743 < clt 7924 ≤ cle 7925 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltwlin 7857 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: lelttrdi 8315 lediv12a 8780 btwnapz 9312 rpgecl 9609 fznatpl1 10001 elfz1b 10015 exbtwnzlemstep 10173 ceiqle 10238 modqabs 10282 mulp1mod1 10290 seq3f1olemqsumk 10424 expgt1 10483 leexp2a 10498 bernneq3 10566 expnbnd 10567 nn0opthlem2d 10623 cvg1nlemres 10913 resqrexlemlo 10941 resqrexlemnmsq 10945 resqrexlemga 10951 abssubap0 11018 icodiamlt 11108 rpmaxcl 11151 reccn2ap 11240 divcnv 11424 cvgratnnlembern 11450 cvgratnnlemabsle 11454 fprodntrivap 11511 efcllemp 11585 sin01bnd 11684 cos01bnd 11685 sin01gt0 11688 cos12dec 11694 eirraplem 11703 dvdslelemd 11766 dvdsbnd 11874 isprm5 12053 znnen 12274 nninfdclemp1 12328 cnopnap 13141 dedekindeulemlu 13146 suplociccreex 13149 dedekindicclemlu 13155 dedekindicc 13158 ivthinclemlopn 13161 limcimolemlt 13180 limccnp2lem 13192 coseq00topi 13303 cosordlem 13317 logdivlti 13349 |
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