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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 2824 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1cmn 20588 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ CMnd |
3 | 1 | xrge0subm 20589 | . . 3 ⊢ (0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
4 | xrex 12389 | . . . . . . 7 ⊢ ℝ* ∈ V | |
5 | 4 | difexi 5235 | . . . . . 6 ⊢ (ℝ* ∖ {-∞}) ∈ V |
6 | difss 4111 | . . . . . . . . 9 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
7 | xrsbas 20564 | . . . . . . . . . 10 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
8 | 1, 7 | ressbas2 16558 | . . . . . . . . 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 17977 | . . . . . . 7 ⊢ ((0[,]+∞) ∈ (SubMnd‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) → (0[,]+∞) ⊆ (ℝ* ∖ {-∞})) |
11 | 3, 10 | ax-mp 5 | . . . . . 6 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
12 | ressabs 16566 | . . . . . 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 2833 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
15 | 14 | submmnd 17981 | . . 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 18960 | . 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 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 0cc0 10540 +∞cpnf 10675 -∞cmnf 10676 ℝ*cxr 10677 [,]cicc 12744 Basecbs 16486 ↾s cress 16487 ℝ*𝑠cxrs 16776 Mndcmnd 17914 SubMndcsubmnd 17958 CMndccmn 18909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-xadd 12511 df-icc 12748 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-tset 16587 df-ple 16588 df-ds 16590 df-0g 16718 df-xrs 16778 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-cmn 18911 |
This theorem is referenced by: xrge0gsumle 23444 xrge0tsms 23445 xrge00 30677 xrge0tsmsd 30696 xrge0omnd 30716 xrge0slmod 30921 xrge0iifmhm 31186 xrge0tmdALT 31193 esumcl 31293 esumgsum 31308 esum0 31312 esumf1o 31313 esumsplit 31316 esumadd 31320 gsumesum 31322 esumlub 31323 esumaddf 31324 esumsnf 31327 esumss 31335 esumpfinval 31338 esumpfinvalf 31339 esumcocn 31343 esum2d 31356 sitmcl 31613 gsumge0cl 42660 sge0tsms 42669 |
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