Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0omnd | Structured version Visualization version GIF version |
Description: The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.) |
Ref | Expression |
---|---|
xrge0omnd | ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0cmn 19788 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
2 | cmnmnd 18208 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
4 | ovex 6678 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ V | |
5 | xrge0base 29685 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | xrge0le 29688 | . . . 4 ⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | iccssxr 12256 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
8 | 7 | sseli 3599 | . . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ ℝ*) |
9 | xrleid 11983 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ≤ 𝑥) |
11 | 7 | sseli 3599 | . . . . 5 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*) |
12 | xrletri3 11985 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | |
13 | 12 | biimprd 238 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
14 | 8, 11, 13 | syl2an 494 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
15 | 7 | sseli 3599 | . . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ*) |
16 | xrletr 11989 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | |
17 | 8, 11, 15, 16 | syl3an 1368 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
18 | 4, 5, 6, 10, 14, 17 | isposi 16956 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset |
19 | xrletri 11984 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | |
20 | 8, 11, 19 | syl2an 494 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
21 | 20 | rgen2a 2977 | . . 3 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) |
22 | 5, 6 | istos 17035 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset ↔ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
23 | 18, 21, 22 | mpbir2an 955 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset |
24 | xleadd1a 12083 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) ∧ 𝑥 ≤ 𝑦) → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) | |
25 | 24 | ex 450 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
26 | 8, 11, 15, 25 | syl3an 1368 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
27 | 26 | rgen3 2976 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)∀𝑧 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) |
28 | xrge0plusg 29687 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 5, 28, 6 | isomnd 29701 | . 2 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd ↔ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)∀𝑧 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)))) |
30 | 3, 23, 27, 29 | mpbir3an 1244 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 +𝑒 cxad 11944 [,]cicc 12178 ↾s cress 15858 ℝ*𝑠cxrs 16160 Posetcpo 16940 Tosetctos 17033 Mndcmnd 17294 CMndccmn 18193 oMndcomnd 29697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-xadd 11947 df-icc 12182 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-xrs 16162 df-poset 16946 df-toset 17034 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-cmn 18195 df-omnd 29699 |
This theorem is referenced by: (None) |
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