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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 12530. (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 12530 | . . 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 2105 class class class wbr 5057 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 |
This theorem is referenced by: qextltlem 12583 ioounsn 12851 snunioc 12854 pcadd2 16214 xblss2ps 22938 xblss2 22939 blhalf 22942 blssps 22961 blss 22962 blcvx 23333 tgqioo 23335 metdcnlem 23371 ioorcl2 24100 volivth 24135 itg2monolem2 24279 itg2cnlem2 24290 dvferm1lem 24508 dvferm2lem 24510 dvferm 24512 dvivthlem1 24532 lhop2 24539 radcnvle 24935 difioo 30431 heicant 34808 ftc1anclem7 34854 supxrgere 41477 suplesup 41483 infrpge 41495 xralrple2 41498 xrralrecnnle 41529 xrralrecnnge 41538 supxrunb3 41548 unb2ltle 41565 xrpnf 41638 snunioo1 41664 iccdifprioo 41668 iccdificc 41691 lptioo1 41789 limsupub 41861 limsuppnflem 41867 limsupre3lem 41889 xlimmnfvlem1 41989 xlimpnfvlem1 41993 fourierdlem46 42314 fourierdlem48 42316 fourierdlem49 42317 fourierdlem74 42342 fourierdlem75 42343 fourierdlem113 42381 ioorrnopnxrlem 42468 salexct2 42499 sge0iunmptlemre 42574 sge0rpcpnf 42580 sge0xaddlem1 42592 meaiuninc3v 42643 ovnsubaddlem1 42729 hoidmv1le 42753 hoidmvlelem5 42758 ovolval4lem1 42808 ovolval5lem1 42811 pimltmnf2 42856 pimgtpnf2 42862 preimageiingt 42875 preimaleiinlt 42876 iccpartleu 43465 iccpartgel 43466 |
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