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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 13194 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . 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 2105 class class class wbr 5147 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 |
This theorem is referenced by: xlt2add 13298 ixxub 13404 elioc2 13446 elicc2 13448 limsupgre 15513 xrsdsreclblem 21447 mnfnei 23244 blgt0 24424 xblss2ps 24426 xblss2 24427 metustexhalf 24584 tgioo 24831 blcvx 24833 xrge0tsms 24869 metdcnlem 24871 metdscnlem 24890 ioombl 25613 uniioombllem1 25629 dvferm2lem 26038 dvlip2 26048 ftc1a 26092 coe1mul3 26152 ply1remlem 26218 idomrootle 26226 pserulm 26479 isblo3i 30829 xrge0infss 32770 iocinioc2 32787 xrge0tsmsd 33047 deg1addlt 33599 q1pvsca 33603 ply1degltdimlem 33649 ply1degltdim 33650 rtelextdg2lem 33731 sibfinima 34320 heicant 37641 itg2gt0cn 37661 ftc1anclem7 37685 ftc1anc 37687 dvrelog3 42046 aks6d1c5lem3 42118 aks6d1c6lem1 42151 aks6d1c6lem3 42153 supxrgelem 45286 supxrge 45287 xralrple2 45303 infxr 45316 infleinflem2 45320 xrralrecnnle 45332 unb2ltle 45364 eliocre 45461 iocopn 45472 ge0lere 45484 iccdificc 45491 limsupre 45596 limsuppnflem 45665 limsupre3lem 45687 limsupub2 45767 xlimmnfv 45789 fourierdlem27 46089 sge0isum 46382 meassre 46432 meaiuninclem 46435 omessre 46465 omeiunltfirp 46474 sge0hsphoire 46544 hoidmv1lelem1 46546 hoidmv1lelem2 46547 hoidmv1lelem3 46548 hoidmvlelem1 46550 hoidmvlelem4 46553 pimiooltgt 46665 pimincfltioc 46671 preimaleiinlt 46676 fsupdm 46797 |
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