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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 2825 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1cmn 20146 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd |
3 | 1 | xrge0subm 20147 | . . 3 ⊢ (0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
4 | xrex 12109 | . . . . . . 7 ⊢ ℝ* ∈ V | |
5 | difss 3964 | . . . . . . 7 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
6 | 4, 5 | ssexi 5028 | . . . . . 6 ⊢ (ℝ* ∖ {-∞}) ∈ V |
7 | xrsbas 20122 | . . . . . . . . . 10 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
8 | 1, 7 | ressbas2 16294 | . . . . . . . . 9 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞})))) |
9 | 5, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
10 | 9 | submss 17703 | . . . . . . 7 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
12 | ressabs 16303 | . . . . . 6 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))) | |
13 | 6, 11, 12 | mp2an 685 | . . . . 5 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) |
14 | 13 | eqcomi 2834 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
15 | 14 | submmnd 17707 | . . 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 18595 | . 2 ⊢ (((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | 2, 16, 17 | mp2an 685 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 {csn 4397 ‘cfv 6123 (class class class)co 6905 0cc0 10252 +∞cpnf 10388 -∞cmnf 10389 ℝ*cxr 10390 [,]cicc 12466 Basecbs 16222 ↾s cress 16223 ℝ*𝑠cxrs 16513 Mndcmnd 17647 SubMndcsubmnd 17687 CMndccmn 18546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-xadd 12233 df-icc 12470 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-tset 16324 df-ple 16325 df-ds 16327 df-0g 16455 df-xrs 16515 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-cmn 18548 |
This theorem is referenced by: xrge0gsumle 23006 xrge0tsms 23007 xrge00 30231 xrge0omnd 30256 xrge0tsmsd 30330 xrge0slmod 30389 xrge0iifmhm 30530 xrge0tmdOLD 30536 esumcl 30637 esumgsum 30652 esum0 30656 esumf1o 30657 esumsplit 30660 esumadd 30664 gsumesum 30666 esumlub 30667 esumaddf 30668 esumsnf 30671 esumss 30679 esumpfinval 30682 esumpfinvalf 30683 esumcocn 30687 esum2d 30700 sitmcl 30958 gsumge0cl 41379 sge0tsms 41388 |
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