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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0omnd | Structured version Visualization version GIF version |
Description: The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.) |
Ref | Expression |
---|---|
xrge0omnd | β’ (β*π βΎs (0[,]+β)) β oMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0cmn 20855 | . . 3 β’ (β*π βΎs (0[,]+β)) β CMnd | |
2 | cmnmnd 19584 | . . 3 β’ ((β*π βΎs (0[,]+β)) β CMnd β (β*π βΎs (0[,]+β)) β Mnd) | |
3 | 1, 2 | ax-mp 5 | . 2 β’ (β*π βΎs (0[,]+β)) β Mnd |
4 | ovex 7391 | . . . 4 β’ (β*π βΎs (0[,]+β)) β V | |
5 | xrge0base 31925 | . . . 4 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
6 | xrge0le 31928 | . . . 4 β’ β€ = (leβ(β*π βΎs (0[,]+β))) | |
7 | eliccxr 13358 | . . . . 5 β’ (π₯ β (0[,]+β) β π₯ β β*) | |
8 | 7 | xrleidd 13077 | . . . 4 β’ (π₯ β (0[,]+β) β π₯ β€ π₯) |
9 | eliccxr 13358 | . . . . 5 β’ (π¦ β (0[,]+β) β π¦ β β*) | |
10 | xrletri3 13079 | . . . . . 6 β’ ((π₯ β β* β§ π¦ β β*) β (π₯ = π¦ β (π₯ β€ π¦ β§ π¦ β€ π₯))) | |
11 | 10 | biimprd 248 | . . . . 5 β’ ((π₯ β β* β§ π¦ β β*) β ((π₯ β€ π¦ β§ π¦ β€ π₯) β π₯ = π¦)) |
12 | 7, 9, 11 | syl2an 597 | . . . 4 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β ((π₯ β€ π¦ β§ π¦ β€ π₯) β π₯ = π¦)) |
13 | eliccxr 13358 | . . . . 5 β’ (π§ β (0[,]+β) β π§ β β*) | |
14 | xrletr 13083 | . . . . 5 β’ ((π₯ β β* β§ π¦ β β* β§ π§ β β*) β ((π₯ β€ π¦ β§ π¦ β€ π§) β π₯ β€ π§)) | |
15 | 7, 9, 13, 14 | syl3an 1161 | . . . 4 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β) β§ π§ β (0[,]+β)) β ((π₯ β€ π¦ β§ π¦ β€ π§) β π₯ β€ π§)) |
16 | 4, 5, 6, 8, 12, 15 | isposi 18218 | . . 3 β’ (β*π βΎs (0[,]+β)) β Poset |
17 | xrletri 13078 | . . . . 5 β’ ((π₯ β β* β§ π¦ β β*) β (π₯ β€ π¦ β¨ π¦ β€ π₯)) | |
18 | 7, 9, 17 | syl2an 597 | . . . 4 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (π₯ β€ π¦ β¨ π¦ β€ π₯)) |
19 | 18 | rgen2 3191 | . . 3 β’ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(π₯ β€ π¦ β¨ π¦ β€ π₯) |
20 | 5, 6 | istos 18312 | . . 3 β’ ((β*π βΎs (0[,]+β)) β Toset β ((β*π βΎs (0[,]+β)) β Poset β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(π₯ β€ π¦ β¨ π¦ β€ π₯))) |
21 | 16, 19, 20 | mpbir2an 710 | . 2 β’ (β*π βΎs (0[,]+β)) β Toset |
22 | xleadd1a 13178 | . . . . 5 β’ (((π₯ β β* β§ π¦ β β* β§ π§ β β*) β§ π₯ β€ π¦) β (π₯ +π π§) β€ (π¦ +π π§)) | |
23 | 22 | ex 414 | . . . 4 β’ ((π₯ β β* β§ π¦ β β* β§ π§ β β*) β (π₯ β€ π¦ β (π₯ +π π§) β€ (π¦ +π π§))) |
24 | 7, 9, 13, 23 | syl3an 1161 | . . 3 β’ ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β) β§ π§ β (0[,]+β)) β (π₯ β€ π¦ β (π₯ +π π§) β€ (π¦ +π π§))) |
25 | 24 | rgen3 3196 | . 2 β’ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)βπ§ β (0[,]+β)(π₯ β€ π¦ β (π₯ +π π§) β€ (π¦ +π π§)) |
26 | xrge0plusg 31927 | . . 3 β’ +π = (+gβ(β*π βΎs (0[,]+β))) | |
27 | 5, 26, 6 | isomnd 31958 | . 2 β’ ((β*π βΎs (0[,]+β)) β oMnd β ((β*π βΎs (0[,]+β)) β Mnd β§ (β*π βΎs (0[,]+β)) β Toset β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)βπ§ β (0[,]+β)(π₯ β€ π¦ β (π₯ +π π§) β€ (π¦ +π π§)))) |
28 | 3, 21, 25, 27 | mpbir3an 1342 | 1 β’ (β*π βΎs (0[,]+β)) β oMnd |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 class class class wbr 5106 (class class class)co 7358 0cc0 11056 +βcpnf 11191 β*cxr 11193 β€ cle 11195 +π cxad 13036 [,]cicc 13273 βΎs cress 17117 β*π cxrs 17387 Posetcpo 18201 Tosetctos 18310 Mndcmnd 18561 CMndccmn 19567 oMndcomnd 31954 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-xadd 13039 df-icc 13277 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-tset 17157 df-ple 17158 df-ds 17160 df-0g 17328 df-xrs 17389 df-poset 18207 df-toset 18311 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-cmn 19569 df-omnd 31956 |
This theorem is referenced by: (None) |
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