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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 13200 | . . 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 ≤ 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: xaddge0 13300 ixxub 13408 ixxlb 13409 limsupval2 15516 0ram 17058 xpsdsval 24391 xblss2ps 24411 xblss2 24412 comet 24526 stdbdxmet 24528 nmoleub 24752 metnrmlem1 24881 nmoleub2lem 25147 ovollb2lem 25523 ovoliunlem2 25538 ovolscalem1 25548 ovolicc1 25551 ovolicc2lem4 25555 voliunlem2 25586 uniioombllem3 25620 itg2uba 25778 itg2lea 25779 itg2split 25784 itg2monolem3 25787 itg2gt0 25795 lhop1lem 26052 dvfsumlem2 26067 dvfsumlem2OLD 26068 dvfsumlem3 26069 dvfsumlem4 26070 deg1addle2 26141 deg1sublt 26149 nmooge0 30786 ply1degltlss 33617 metideq 33892 measiun 34219 omssubadd 34302 carsgclctunlem2 34321 mblfinlem1 37664 ismblfin 37668 ftc1anclem8 37707 ftc1anc 37708 aks6d1c6lem2 42172 aks6d1c6lem3 42173 unitscyglem5 42200 hbtlem2 43136 idomodle 43203 xle2addd 45347 xralrple2 45365 infleinflem1 45381 xralrple4 45384 xralrple3 45385 suplesup2 45387 infleinf2 45425 infxrlesupxr 45447 inficc 45547 limsupequzlem 45737 limsupvaluz2 45753 supcnvlimsup 45755 liminfval2 45783 liminflelimsuplem 45790 limsupgtlem 45792 fourierdlem1 46123 sge0cl 46396 sge0lefi 46413 sge0iunmptlemre 46430 sge0isum 46442 omeunle 46531 omeiunle 46532 caratheodorylem2 46542 hoicvrrex 46571 ovnsubaddlem1 46585 ovolval5lem1 46667 pimdecfgtioo 46732 pimincfltioo 46733 |
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