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| Mirrors > Home > MPE Home > Th. List > xrlelttrd | Structured version Visualization version GIF version | ||
| Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| Ref | Expression |
|---|---|
| xrlttrd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| xrlttrd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| xrlttrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
| xrlelttrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| xrlelttrd.5 | ⊢ (𝜑 → 𝐵 < 𝐶) |
| Ref | Expression |
|---|---|
| xrlelttrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrlelttrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 2 | xrlelttrd.5 | . 2 ⊢ (𝜑 → 𝐵 < 𝐶) | |
| 3 | xrlttrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 4 | xrlttrd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 5 | xrlttrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
| 6 | xrlelttr 13198 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
| 7 | 3, 4, 5, 6 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
| 8 | 1, 2, 7 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: xlt2add 13302 ixxub 13408 elioc2 13450 elicc2 13452 limsupgre 15517 xrsdsreclblem 21430 mnfnei 23229 blgt0 24409 xblss2ps 24411 xblss2 24412 metustexhalf 24569 tgioo 24817 blcvx 24819 xrge0tsms 24856 metdcnlem 24858 metdscnlem 24877 ioombl 25600 uniioombllem1 25616 dvferm2lem 26024 dvlip2 26034 ftc1a 26078 coe1mul3 26138 ply1remlem 26204 idomrootle 26212 pserulm 26465 isblo3i 30820 xrge0infss 32764 iocinioc2 32781 xrge0tsmsd 33065 deg1addlt 33620 q1pvsca 33624 ply1degltdimlem 33673 ply1degltdim 33674 rtelextdg2lem 33767 sibfinima 34341 heicant 37662 itg2gt0cn 37682 ftc1anclem7 37706 ftc1anc 37708 dvrelog3 42066 aks6d1c5lem3 42138 aks6d1c6lem1 42171 aks6d1c6lem3 42173 supxrgelem 45348 supxrge 45349 xralrple2 45365 infxr 45378 infleinflem2 45382 xrralrecnnle 45394 unb2ltle 45426 eliocre 45522 iocopn 45533 ge0lere 45545 iccdificc 45552 limsupre 45656 limsuppnflem 45725 limsupre3lem 45747 limsupub2 45827 xlimmnfv 45849 fourierdlem27 46149 sge0isum 46442 meassre 46492 meaiuninclem 46495 omessre 46525 omeiunltfirp 46534 sge0hsphoire 46604 hoidmv1lelem1 46606 hoidmv1lelem2 46607 hoidmv1lelem3 46608 hoidmvlelem1 46610 hoidmvlelem4 46613 pimiooltgt 46725 pimincfltioc 46731 preimaleiinlt 46736 fsupdm 46857 |
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