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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 21426 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
| 2 | cmnmnd 19815 | . . 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 7464 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ V | |
| 5 | xrge0base 33016 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 6 | xrge0le 33019 | . . . 4 ⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 7 | eliccxr 13475 | . . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ ℝ*) | |
| 8 | 7 | xrleidd 13194 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ≤ 𝑥) |
| 9 | eliccxr 13475 | . . . . 5 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*) | |
| 10 | xrletri3 13196 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | |
| 11 | 10 | biimprd 248 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
| 12 | 7, 9, 11 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
| 13 | eliccxr 13475 | . . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ*) | |
| 14 | xrletr 13200 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | |
| 15 | 7, 9, 13, 14 | syl3an 1161 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
| 16 | 4, 5, 6, 8, 12, 15 | isposi 18369 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset |
| 17 | xrletri 13195 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | |
| 18 | 7, 9, 17 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
| 19 | 18 | rgen2 3199 | . . 3 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) |
| 20 | 5, 6 | istos 18463 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset ↔ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Poset ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
| 21 | 16, 19, 20 | mpbir2an 711 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Toset |
| 22 | xleadd1a 13295 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) ∧ 𝑥 ≤ 𝑦) → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) | |
| 23 | 22 | ex 412 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
| 24 | 7, 9, 13, 23 | syl3an 1161 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞) ∧ 𝑧 ∈ (0[,]+∞)) → (𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧))) |
| 25 | 24 | rgen3 3204 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)∀𝑧 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝑥 +𝑒 𝑧) ≤ (𝑦 +𝑒 𝑧)) |
| 26 | xrge0plusg 33018 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 27 | 5, 26, 6 | isomnd 33078 | . 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 1342 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 +𝑒 cxad 13152 [,]cicc 13390 ↾s cress 17274 ℝ*𝑠cxrs 17545 Posetcpo 18353 Tosetctos 18461 Mndcmnd 18747 CMndccmn 19798 oMndcomnd 33074 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-xadd 13155 df-icc 13394 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-xrs 17547 df-poset 18359 df-toset 18462 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-cmn 19800 df-omnd 33076 |
| This theorem is referenced by: (None) |
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