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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 7853 | . . 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 < clt 7800 ≤ 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: lelttrdi 8188 lediv12a 8652 btwnapz 9181 rpgecl 9470 fznatpl1 9856 elfz1b 9870 exbtwnzlemstep 10025 ceiqle 10086 modqabs 10130 mulp1mod1 10138 seq3f1olemqsumk 10272 expgt1 10331 leexp2a 10346 bernneq3 10414 expnbnd 10415 nn0opthlem2d 10467 cvg1nlemres 10757 resqrexlemlo 10785 resqrexlemnmsq 10789 resqrexlemga 10795 abssubap0 10862 icodiamlt 10952 rpmaxcl 10995 reccn2ap 11082 divcnv 11266 cvgratnnlembern 11292 cvgratnnlemabsle 11296 efcllemp 11364 sin01bnd 11464 cos01bnd 11465 sin01gt0 11468 cos12dec 11474 eirraplem 11483 dvdslelemd 11541 dvdsbnd 11645 znnen 11911 cnopnap 12763 dedekindeulemlu 12768 suplociccreex 12771 dedekindicclemlu 12777 dedekindicc 12780 ivthinclemlopn 12783 limcimolemlt 12802 limccnp2lem 12814 coseq00topi 12916 cosordlem 12930 |
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