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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 2821 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1cmn 20579 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd |
3 | 1 | xrge0subm 20580 | . . 3 ⊢ (0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
4 | xrex 12380 | . . . . . . 7 ⊢ ℝ* ∈ V | |
5 | 4 | difexi 5225 | . . . . . 6 ⊢ (ℝ* ∖ {-∞}) ∈ V |
6 | difss 4108 | . . . . . . . . 9 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
7 | xrsbas 20555 | . . . . . . . . . 10 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
8 | 1, 7 | ressbas2 16549 | . . . . . . . . 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 17968 | . . . . . . 7 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
12 | ressabs 16557 | . . . . . 6 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))) | |
13 | 5, 11, 12 | mp2an 690 | . . . . 5 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) |
14 | 13 | eqcomi 2830 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
15 | 14 | submmnd 17972 | . . 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 18951 | . 2 ⊢ (((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | 2, 16, 17 | mp2an 690 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 ⊆ wss 3936 {csn 4561 ‘cfv 6350 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 -∞cmnf 10667 ℝ*cxr 10668 [,]cicc 12735 Basecbs 16477 ↾s cress 16478 ℝ*𝑠cxrs 16767 Mndcmnd 17905 SubMndcsubmnd 17949 CMndccmn 18900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-xadd 12502 df-icc 12739 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-xrs 16769 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-cmn 18902 |
This theorem is referenced by: xrge0gsumle 23435 xrge0tsms 23436 xrge00 30668 xrge0tsmsd 30687 xrge0omnd 30707 xrge0slmod 30912 xrge0iifmhm 31177 xrge0tmdALT 31184 esumcl 31284 esumgsum 31299 esum0 31303 esumf1o 31304 esumsplit 31307 esumadd 31311 gsumesum 31313 esumlub 31314 esumaddf 31315 esumsnf 31318 esumss 31326 esumpfinval 31329 esumpfinvalf 31330 esumcocn 31334 esum2d 31347 sitmcl 31604 gsumge0cl 42646 sge0tsms 42655 |
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