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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 2734 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1cmn 21441 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd |
3 | 1 | xrge0subm 21442 | . . 3 ⊢ (0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
4 | xrex 13026 | . . . . . . 7 ⊢ ℝ* ∈ V | |
5 | 4 | difexi 5335 | . . . . . 6 ⊢ (ℝ* ∖ {-∞}) ∈ V |
6 | difss 4145 | . . . . . . . . 9 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
7 | xrsbas 21413 | . . . . . . . . . 10 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
8 | 1, 7 | ressbas2 17282 | . . . . . . . . 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 18834 | . . . . . . 7 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
12 | ressabs 17294 | . . . . . 6 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))) | |
13 | 5, 11, 12 | mp2an 692 | . . . . 5 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) |
14 | 13 | eqcomi 2743 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
15 | 14 | submmnd 18838 | . . 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 19869 | . 2 ⊢ (((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | 2, 16, 17 | mp2an 692 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 ‘cfv 6562 (class class class)co 7430 0cc0 11152 +∞cpnf 11289 -∞cmnf 11290 ℝ*cxr 11291 [,]cicc 13386 Basecbs 17244 ↾s cress 17273 ℝ*𝑠cxrs 17546 Mndcmnd 18759 SubMndcsubmnd 18807 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-xadd 13152 df-icc 13390 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17487 df-xrs 17548 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-cmn 19814 |
This theorem is referenced by: xrge0gsumle 24868 xrge0tsms 24869 xrge00 32999 xrge0tsmsd 33047 xrge0omnd 33070 xrge0slmod 33355 xrge0iifmhm 33899 xrge0tmdALT 33906 esumcl 34010 esumgsum 34025 esum0 34029 esumf1o 34030 esumsplit 34033 esumadd 34037 gsumesum 34039 esumlub 34040 esumaddf 34041 esumsnf 34044 esumss 34052 esumpfinval 34055 esumpfinvalf 34056 esumcocn 34060 esum2d 34073 sitmcl 34332 gsumge0cl 46326 sge0tsms 46335 |
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