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| Mirrors > Home > MPE Home > Th. List > xrltletrd | 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*) |
| xrltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
| xrltletrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
| Ref | Expression |
|---|---|
| xrltletrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 2 | xrltletrd.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 | xrltletr 13199 | . . 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 xadddi2 13339 supxrre 13369 infxrre 13378 ixxlb 13409 elicore 13439 elico2 13451 elicc2 13452 caucvgrlem 15709 isnzr2hash 20519 xrsdsreclblem 21430 xblss2ps 24411 xblss2 24412 tgioo 24817 xrge0tsms 24856 xrhmeo 24977 ovoliunlem1 25537 ovoliun 25540 ioombl1lem2 25594 vitalilem4 25646 itg2monolem2 25786 itg2gt0 25795 dvferm1lem 26022 dvferm2lem 26024 lhop1lem 26052 pserdvlem2 26472 abelthlem3 26477 logtayl 26702 xrge0tsmsd 33065 ply1degltdimlem 33673 esum2d 34094 usgrcyclgt2v 35136 relowlssretop 37364 itg2gt0cn 37682 areacirclem5 37719 aks6d1c6lem3 42173 aks6d1c7lem2 42182 xrge0nemnfd 45343 supxrgere 45344 supxrgelem 45348 infrpge 45362 xrralrecnnge 45401 supxrunb3 45410 icoopn 45538 limsupre 45656 limsupre3lem 45747 xlimpnfv 45853 fourierdlem27 46149 fourierdlem87 46208 gsumge0cl 46386 sge0pr 46409 sge0ssre 46412 sge0xaddlem1 46448 meaiuninc3v 46499 pimiooltgt 46725 pimdecfgtioc 46730 preimageiingt 46735 finfdm 46861 |
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