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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 483 | . . . . 5 β’ ((π₯ β β* β§ 0 β€ π₯) β π₯ β β*) | |
2 | ge0nemnf 13020 | . . . . 5 β’ ((π₯ β β* β§ 0 β€ π₯) β π₯ β -β) | |
3 | 1, 2 | jca 512 | . . . 4 β’ ((π₯ β β* β§ 0 β€ π₯) β (π₯ β β* β§ π₯ β -β)) |
4 | elxrge0 13302 | . . . 4 β’ (π₯ β (0[,]+β) β (π₯ β β* β§ 0 β€ π₯)) | |
5 | eldifsn 4745 | . . . 4 β’ (π₯ β (β* β {-β}) β (π₯ β β* β§ π₯ β -β)) | |
6 | 3, 4, 5 | 3imtr4i 291 | . . 3 β’ (π₯ β (0[,]+β) β π₯ β (β* β {-β})) |
7 | 6 | ssriv 3946 | . 2 β’ (0[,]+β) β (β* β {-β}) |
8 | 0e0iccpnf 13304 | . 2 β’ 0 β (0[,]+β) | |
9 | ge0xaddcl 13307 | . . 3 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (π₯ +π π¦) β (0[,]+β)) | |
10 | 9 | rgen2 3192 | . 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 4089 | . . . . 5 β’ (β* β {-β}) β β* | |
14 | xrsbas 20736 | . . . . . 6 β’ β* = (Baseββ*π ) | |
15 | 11, 14 | ressbas2 17054 | . . . . 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 12840 | . . . . . 6 β’ β* β V | |
19 | 18 | difexi 5283 | . . . . 5 β’ (β* β {-β}) β V |
20 | xrsadd 20737 | . . . . . 6 β’ +π = (+gββ*π ) | |
21 | 11, 20 | ressplusg 17105 | . . . . 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 18548 | . . 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 1341 | 1 β’ (0[,]+β) β (SubMndβπ ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2941 βwral 3062 Vcvv 3443 β cdif 3905 β wss 3908 {csn 4584 class class class wbr 5103 βcfv 6491 (class class class)co 7349 0cc0 10984 +βcpnf 11119 -βcmnf 11120 β*cxr 11121 β€ cle 11123 +π cxad 12959 [,]cicc 13195 Basecbs 17017 βΎs cress 17046 +gcplusg 17067 β*π cxrs 17316 Mndcmnd 18490 SubMndcsubmnd 18534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-xadd 12962 df-icc 13199 df-fz 13353 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-tset 17086 df-ple 17087 df-ds 17089 df-0g 17257 df-xrs 17318 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-submnd 18536 |
This theorem is referenced by: xrge0cmn 20762 xrge0gsumle 24118 xrge0tsms 24119 xrge0tsmsd 31693 |
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