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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 7724 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1184 | . 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 427 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1448 class class class wbr 3875 ℝcr 7499 < clt 7672 ≤ cle 7673 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-pre-ltwlin 7608 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-cnv 4485 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 |
This theorem is referenced by: lelttrdi 8055 lediv12a 8510 btwnapz 9033 rpgecl 9319 fznatpl1 9697 elfz1b 9711 exbtwnzlemstep 9866 ceiqle 9927 modqabs 9971 mulp1mod1 9979 seq3f1olemqsumk 10113 expgt1 10172 leexp2a 10187 bernneq3 10255 expnbnd 10256 nn0opthlem2d 10308 cvg1nlemres 10597 resqrexlemlo 10625 resqrexlemnmsq 10629 resqrexlemga 10635 abssubap0 10702 icodiamlt 10792 reccn2ap 10921 divcnv 11105 cvgratnnlembern 11131 cvgratnnlemabsle 11135 efcllemp 11162 sin01bnd 11262 cos01bnd 11263 sin01gt0 11266 eirraplem 11278 dvdslelemd 11336 dvdsbnd 11440 znnen 11703 limcimolemlt 12513 |
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