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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 2726 | . . 3 β’ (β*π βΎs (β* β {-β})) = (β*π βΎs (β* β {-β})) | |
2 | 1 | xrs1cmn 21300 | . 2 β’ (β*π βΎs (β* β {-β})) β CMnd |
3 | 1 | xrge0subm 21301 | . . 3 β’ (0[,]+β) β (SubMndβ(β*π βΎs (β* β {-β}))) |
4 | xrex 12975 | . . . . . . 7 β’ β* β V | |
5 | 4 | difexi 5321 | . . . . . 6 β’ (β* β {-β}) β V |
6 | difss 4126 | . . . . . . . . 9 β’ (β* β {-β}) β β* | |
7 | xrsbas 21272 | . . . . . . . . . 10 β’ β* = (Baseββ*π ) | |
8 | 1, 7 | ressbas2 17191 | . . . . . . . . 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 18734 | . . . . . . 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 17203 | . . . . . 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 689 | . . . . 5 β’ ((β*π βΎs (β* β {-β})) βΎs (0[,]+β)) = (β*π βΎs (0[,]+β)) |
14 | 13 | eqcomi 2735 | . . . 4 β’ (β*π βΎs (0[,]+β)) = ((β*π βΎs (β* β {-β})) βΎs (0[,]+β)) |
15 | 14 | submmnd 18738 | . . 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 19757 | . 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 689 | 1 β’ (β*π βΎs (0[,]+β)) β CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 Vcvv 3468 β cdif 3940 β wss 3943 {csn 4623 βcfv 6537 (class class class)co 7405 0cc0 11112 +βcpnf 11249 -βcmnf 11250 β*cxr 11251 [,]cicc 13333 Basecbs 17153 βΎs cress 17182 β*π cxrs 17455 Mndcmnd 18667 SubMndcsubmnd 18712 CMndccmn 19700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-xadd 13099 df-icc 13337 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-tset 17225 df-ple 17226 df-ds 17228 df-0g 17396 df-xrs 17457 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-cmn 19702 |
This theorem is referenced by: xrge0gsumle 24704 xrge0tsms 24705 xrge00 32690 xrge0tsmsd 32715 xrge0omnd 32735 xrge0slmod 32966 xrge0iifmhm 33449 xrge0tmdALT 33456 esumcl 33558 esumgsum 33573 esum0 33577 esumf1o 33578 esumsplit 33581 esumadd 33585 gsumesum 33587 esumlub 33588 esumaddf 33589 esumsnf 33592 esumss 33600 esumpfinval 33603 esumpfinvalf 33604 esumcocn 33608 esum2d 33621 sitmcl 33880 gsumge0cl 45659 sge0tsms 45668 |
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