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Mirrors > Home > MPE Home > Th. List > xrge0cmn | Structured version Visualization version GIF version |
Description: The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
xrge0cmn | β’ (β*π βΎs (0[,]+β)) β CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 β’ (β*π βΎs (β* β {-β})) = (β*π βΎs (β* β {-β})) | |
2 | 1 | xrs1cmn 20853 | . 2 β’ (β*π βΎs (β* β {-β})) β CMnd |
3 | 1 | xrge0subm 20854 | . . 3 β’ (0[,]+β) β (SubMndβ(β*π βΎs (β* β {-β}))) |
4 | xrex 12919 | . . . . . . 7 β’ β* β V | |
5 | 4 | difexi 5290 | . . . . . 6 β’ (β* β {-β}) β V |
6 | difss 4096 | . . . . . . . . 9 β’ (β* β {-β}) β β* | |
7 | xrsbas 20829 | . . . . . . . . . 10 β’ β* = (Baseββ*π ) | |
8 | 1, 7 | ressbas2 17127 | . . . . . . . . 9 β’ ((β* β {-β}) β β* β (β* β {-β}) = (Baseβ(β*π βΎs (β* β {-β})))) |
9 | 6, 8 | ax-mp 5 | . . . . . . . 8 β’ (β* β {-β}) = (Baseβ(β*π βΎs (β* β {-β}))) |
10 | 9 | submss 18627 | . . . . . . 7 β’ ((0[,]+β) β (SubMndβ(β*π βΎs (β* β {-β}))) β (0[,]+β) β (β* β {-β})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 β’ (0[,]+β) β (β* β {-β}) |
12 | ressabs 17137 | . . . . . 6 β’ (((β* β {-β}) β V β§ (0[,]+β) β (β* β {-β})) β ((β*π βΎs (β* β {-β})) βΎs (0[,]+β)) = (β*π βΎs (0[,]+β))) | |
13 | 5, 11, 12 | mp2an 691 | . . . . 5 β’ ((β*π βΎs (β* β {-β})) βΎs (0[,]+β)) = (β*π βΎs (0[,]+β)) |
14 | 13 | eqcomi 2746 | . . . 4 β’ (β*π βΎs (0[,]+β)) = ((β*π βΎs (β* β {-β})) βΎs (0[,]+β)) |
15 | 14 | submmnd 18631 | . . 3 β’ ((0[,]+β) β (SubMndβ(β*π βΎs (β* β {-β}))) β (β*π βΎs (0[,]+β)) β Mnd) |
16 | 3, 15 | ax-mp 5 | . 2 β’ (β*π βΎs (0[,]+β)) β Mnd |
17 | 14 | subcmn 19622 | . 2 β’ (((β*π βΎs (β* β {-β})) β CMnd β§ (β*π βΎs (0[,]+β)) β Mnd) β (β*π βΎs (0[,]+β)) β CMnd) |
18 | 2, 16, 17 | mp2an 691 | 1 β’ (β*π βΎs (0[,]+β)) β CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 Vcvv 3448 β cdif 3912 β wss 3915 {csn 4591 βcfv 6501 (class class class)co 7362 0cc0 11058 +βcpnf 11193 -βcmnf 11194 β*cxr 11195 [,]cicc 13274 Basecbs 17090 βΎs cress 17119 β*π cxrs 17389 Mndcmnd 18563 SubMndcsubmnd 18607 CMndccmn 19569 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-xadd 13041 df-icc 13278 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-tset 17159 df-ple 17160 df-ds 17162 df-0g 17330 df-xrs 17391 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-cmn 19571 |
This theorem is referenced by: xrge0gsumle 24212 xrge0tsms 24213 xrge00 31919 xrge0tsmsd 31941 xrge0omnd 31961 xrge0slmod 32180 xrge0iifmhm 32560 xrge0tmdALT 32567 esumcl 32669 esumgsum 32684 esum0 32688 esumf1o 32689 esumsplit 32692 esumadd 32696 gsumesum 32698 esumlub 32699 esumaddf 32700 esumsnf 32703 esumss 32711 esumpfinval 32714 esumpfinvalf 32715 esumcocn 32719 esum2d 32732 sitmcl 32991 gsumge0cl 44686 sge0tsms 44695 |
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