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Mirrors > Home > MPE Home > Th. List > xrletrd | 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*) |
xrletrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
xrletrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
xrletrd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrletrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | xrletrd.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 | xrletr 12892 | . . 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 696 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5074 ℝ*cxr 11008 ≤ cle 11010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 |
This theorem is referenced by: xaddge0 12992 ixxub 13100 ixxlb 13101 limsupval2 15189 0ram 16721 xpsdsval 23534 xblss2ps 23554 xblss2 23555 comet 23669 stdbdxmet 23671 nmoleub 23895 metnrmlem1 24022 nmoleub2lem 24277 ovollb2lem 24652 ovoliunlem2 24667 ovolscalem1 24677 ovolicc1 24680 ovolicc2lem4 24684 voliunlem2 24715 uniioombllem3 24749 itg2uba 24908 itg2lea 24909 itg2split 24914 itg2monolem3 24917 itg2gt0 24925 lhop1lem 25177 dvfsumlem2 25191 dvfsumlem3 25192 dvfsumlem4 25193 deg1addle2 25267 deg1sublt 25275 nmooge0 29129 metideq 31843 measiun 32186 omssubadd 32267 carsgclctunlem2 32286 mblfinlem1 35814 ismblfin 35818 ftc1anclem8 35857 ftc1anc 35858 hbtlem2 40949 idomodle 41021 xle2addd 42875 xralrple2 42893 infleinflem1 42909 xralrple4 42912 xralrple3 42913 suplesup2 42915 infleinf2 42954 infxrlesupxr 42976 inficc 43072 limsupequzlem 43263 limsupvaluz2 43279 supcnvlimsup 43281 liminfval2 43309 liminflelimsuplem 43316 limsupgtlem 43318 fourierdlem1 43649 sge0cl 43919 sge0lefi 43936 sge0iunmptlemre 43953 sge0isum 43965 omeunle 44054 omeiunle 44055 caratheodorylem2 44065 hoicvrrex 44094 ovnsubaddlem1 44108 ovolval5lem1 44190 pimdecfgtioo 44254 pimincfltioo 44255 |
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