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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 | ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1cmn 20131 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd |
3 | 1 | xrge0subm 20132 | . . 3 ⊢ (0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
4 | xrex 12374 | . . . . . . 7 ⊢ ℝ* ∈ V | |
5 | 4 | difexi 5196 | . . . . . 6 ⊢ (ℝ* ∖ {-∞}) ∈ V |
6 | difss 4059 | . . . . . . . . 9 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
7 | xrsbas 20107 | . . . . . . . . . 10 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
8 | 1, 7 | ressbas2 16547 | . . . . . . . . 9 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞})))) |
9 | 6, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
10 | 9 | submss 17966 | . . . . . . 7 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
12 | ressabs 16555 | . . . . . 6 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))) | |
13 | 5, 11, 12 | mp2an 691 | . . . . 5 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) |
14 | 13 | eqcomi 2807 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
15 | 14 | submmnd 17970 | . . 3 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) |
16 | 3, 15 | ax-mp 5 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
17 | 14 | subcmn 18950 | . 2 ⊢ (((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | 2, 16, 17 | mp2an 691 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 [,]cicc 12729 Basecbs 16475 ↾s cress 16476 ℝ*𝑠cxrs 16765 Mndcmnd 17903 SubMndcsubmnd 17947 CMndccmn 18898 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-xadd 12496 df-icc 12733 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-xrs 16767 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-cmn 18900 |
This theorem is referenced by: xrge0gsumle 23438 xrge0tsms 23439 xrge00 30720 xrge0tsmsd 30742 xrge0omnd 30762 xrge0slmod 30968 xrge0iifmhm 31292 xrge0tmdALT 31299 esumcl 31399 esumgsum 31414 esum0 31418 esumf1o 31419 esumsplit 31422 esumadd 31426 gsumesum 31428 esumlub 31429 esumaddf 31430 esumsnf 31433 esumss 31441 esumpfinval 31444 esumpfinvalf 31445 esumcocn 31449 esum2d 31462 sitmcl 31719 gsumge0cl 43010 sge0tsms 43019 |
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