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| Mirrors > Home > MPE Home > Th. List > xrltled | Structured version Visualization version GIF version | ||
| Description: 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 13191. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| xrltled.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| xrltled.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| xrltled.altb | ⊢ (𝜑 → 𝐴 < 𝐵) |
| Ref | Expression |
|---|---|
| xrltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrltled.altb | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 2 | xrltled.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 3 | xrltled.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 4 | xrltle 13191 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
| 5 | 2, 3, 4 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
| 6 | 1, 5 | mpd 15 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ 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: qextltlem 13244 ioounsn 13517 snunioc 13520 pcadd2 16928 xblss2ps 24411 xblss2 24412 blhalf 24415 blssps 24434 blss 24435 blcvx 24819 tgqioo 24821 metdcnlem 24858 ioorcl2 25607 volivth 25642 itg2monolem2 25786 itg2cnlem2 25797 dvferm1lem 26022 dvferm2lem 26024 dvferm 26026 dvivthlem1 26047 lhop2 26054 radcnvle 26463 difioo 32784 heicant 37662 ftc1anclem7 37706 supxrgere 45344 suplesup 45350 infrpge 45362 xralrple2 45365 xrralrecnnle 45394 xrralrecnnge 45401 supxrunb3 45410 unb2ltle 45426 xrpnf 45496 snunioo1 45525 iccdifprioo 45529 iccdificc 45552 lptioo1 45647 limsupub 45719 limsuppnflem 45725 limsupre3lem 45747 xlimmnfvlem1 45847 xlimpnfvlem1 45851 fourierdlem46 46167 fourierdlem48 46169 fourierdlem49 46170 fourierdlem74 46195 fourierdlem75 46196 fourierdlem113 46234 ioorrnopnxrlem 46321 salexct2 46354 sge0iunmptlemre 46430 sge0rpcpnf 46436 sge0xaddlem1 46448 meaiuninc3v 46499 ovnsubaddlem1 46585 hoidmv1le 46609 hoidmvlelem5 46614 ovolval4lem1 46664 ovolval5lem1 46667 preimageiingt 46735 preimaleiinlt 46736 fsupdm 46857 finfdm 46861 iccpartleu 47415 iccpartgel 47416 |
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