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 20589 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
2 | cmnmnd 18924 | . . 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 7191 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ V | |
5 | xrge0base 30674 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | xrge0le 30677 | . . . 4 ⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | eliccxr 12826 | . . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ ℝ*) | |
8 | 7 | xrleidd 12548 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ≤ 𝑥) |
9 | eliccxr 12826 | . . . . 5 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*) | |
10 | xrletri3 12550 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | |
11 | 10 | biimprd 250 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
12 | 7, 9, 11 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
13 | eliccxr 12826 | . . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ*) | |
14 | xrletr 12554 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | |
15 | 7, 9, 13, 14 | syl3an 1156 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
16 | 4, 5, 6, 8, 12, 15 | isposi 17568 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset |
17 | xrletri 12549 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | |
18 | 7, 9, 17 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
19 | 18 | rgen2 3205 | . . 3 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) |
20 | 5, 6 | istos 17647 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset ↔ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
21 | 16, 19, 20 | mpbir2an 709 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset |
22 | xleadd1a 12649 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) ∧ 𝑥 ≤ 𝑦) → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) | |
23 | 22 | ex 415 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
24 | 7, 9, 13, 23 | syl3an 1156 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
25 | 24 | rgen3 3206 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)∀𝑧 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) |
26 | xrge0plusg 30676 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
27 | 5, 26, 6 | isomnd 30704 | . 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)))) |
28 | 3, 21, 25, 27 | mpbir3an 1337 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 ≤ cle 10678 +𝑒 cxad 12508 [,]cicc 12744 ↾s cress 16486 ℝ*𝑠cxrs 16775 Posetcpo 17552 Tosetctos 17645 Mndcmnd 17913 CMndccmn 18908 oMndcomnd 30700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-0g 16717 df-xrs 16777 df-poset 17558 df-toset 17646 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-cmn 18910 df-omnd 30702 |
This theorem is referenced by: (None) |
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