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Mirrors > Home > MPE Home > Th. List > xrge0subm | Structured version Visualization version GIF version |
Description: The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrs1mnd.1 | β’ π = (β*π βΎs (β* β {-β})) |
Ref | Expression |
---|---|
xrge0subm | β’ (0[,]+β) β (SubMndβπ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . . 5 β’ ((π₯ β β* β§ 0 β€ π₯) β π₯ β β*) | |
2 | ge0nemnf 13021 | . . . . 5 β’ ((π₯ β β* β§ 0 β€ π₯) β π₯ β -β) | |
3 | 1, 2 | jca 513 | . . . 4 β’ ((π₯ β β* β§ 0 β€ π₯) β (π₯ β β* β§ π₯ β -β)) |
4 | elxrge0 13303 | . . . 4 β’ (π₯ β (0[,]+β) β (π₯ β β* β§ 0 β€ π₯)) | |
5 | eldifsn 4746 | . . . 4 β’ (π₯ β (β* β {-β}) β (π₯ β β* β§ π₯ β -β)) | |
6 | 3, 4, 5 | 3imtr4i 292 | . . 3 β’ (π₯ β (0[,]+β) β π₯ β (β* β {-β})) |
7 | 6 | ssriv 3947 | . 2 β’ (0[,]+β) β (β* β {-β}) |
8 | 0e0iccpnf 13305 | . 2 β’ 0 β (0[,]+β) | |
9 | ge0xaddcl 13308 | . . 3 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (π₯ +π π¦) β (0[,]+β)) | |
10 | 9 | rgen2 3193 | . 2 β’ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(π₯ +π π¦) β (0[,]+β) |
11 | xrs1mnd.1 | . . . 4 β’ π = (β*π βΎs (β* β {-β})) | |
12 | 11 | xrs1mnd 20758 | . . 3 β’ π β Mnd |
13 | difss 4090 | . . . . 5 β’ (β* β {-β}) β β* | |
14 | xrsbas 20736 | . . . . . 6 β’ β* = (Baseββ*π ) | |
15 | 11, 14 | ressbas2 17055 | . . . . 5 β’ ((β* β {-β}) β β* β (β* β {-β}) = (Baseβπ )) |
16 | 13, 15 | ax-mp 5 | . . . 4 β’ (β* β {-β}) = (Baseβπ ) |
17 | 11 | xrs10 20759 | . . . 4 β’ 0 = (0gβπ ) |
18 | xrex 12841 | . . . . . 6 β’ β* β V | |
19 | 18 | difexi 5284 | . . . . 5 β’ (β* β {-β}) β V |
20 | xrsadd 20737 | . . . . . 6 β’ +π = (+gββ*π ) | |
21 | 11, 20 | ressplusg 17106 | . . . . 5 β’ ((β* β {-β}) β V β +π = (+gβπ )) |
22 | 19, 21 | ax-mp 5 | . . . 4 β’ +π = (+gβπ ) |
23 | 16, 17, 22 | issubm 18549 | . . 3 β’ (π β Mnd β ((0[,]+β) β (SubMndβπ ) β ((0[,]+β) β (β* β {-β}) β§ 0 β (0[,]+β) β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(π₯ +π π¦) β (0[,]+β)))) |
24 | 12, 23 | ax-mp 5 | . 2 β’ ((0[,]+β) β (SubMndβπ ) β ((0[,]+β) β (β* β {-β}) β§ 0 β (0[,]+β) β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(π₯ +π π¦) β (0[,]+β))) |
25 | 7, 8, 10, 24 | mpbir3an 1342 | 1 β’ (0[,]+β) β (SubMndβπ ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2942 βwral 3063 Vcvv 3444 β cdif 3906 β wss 3909 {csn 4585 class class class wbr 5104 βcfv 6492 (class class class)co 7350 0cc0 10985 +βcpnf 11120 -βcmnf 11121 β*cxr 11122 β€ cle 11124 +π cxad 12960 [,]cicc 13196 Basecbs 17018 βΎs cress 17047 +gcplusg 17068 β*π cxrs 17317 Mndcmnd 18491 SubMndcsubmnd 18535 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-xadd 12963 df-icc 13200 df-fz 13354 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-tset 17087 df-ple 17088 df-ds 17090 df-0g 17258 df-xrs 17319 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 |
This theorem is referenced by: xrge0cmn 20762 xrge0gsumle 24118 xrge0tsms 24119 xrge0tsmsd 31681 |
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