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Mirrors > Home > MPE Home > Th. List > xrs1mnd | Structured version Visualization version GIF version |
Description: The extended real numbers, restricted to ℝ* ∖ {-∞}, form a monoid - in contrast to the full structure, see xrsmgmdifsgrp 19997. (Contributed by Mario Carneiro, 27-Nov-2014.) |
Ref | Expression |
---|---|
xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
Ref | Expression |
---|---|
xrs1mnd | ⊢ 𝑅 ∈ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3888 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 19976 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 16137 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | mp1i 13 | . . 3 ⊢ (⊤ → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
6 | xrex 12031 | . . . . 5 ⊢ ℝ* ∈ V | |
7 | difexg 4943 | . . . . 5 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
9 | xrsadd 19977 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
10 | 2, 9 | ressplusg 16200 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
11 | 8, 10 | mp1i 13 | . . 3 ⊢ (⊤ → +𝑒 = (+g‘𝑅)) |
12 | eldifsn 4454 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
13 | eldifsn 4454 | . . . . 5 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) ↔ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) | |
14 | xaddcl 12274 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*) | |
15 | 14 | ad2ant2r 741 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ∈ ℝ*) |
16 | xaddnemnf 12271 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ≠ -∞) | |
17 | eldifsn 4454 | . . . . . 6 ⊢ ((𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞}) ↔ ((𝑥 +𝑒 𝑦) ∈ ℝ* ∧ (𝑥 +𝑒 𝑦) ≠ -∞)) | |
18 | 15, 16, 17 | sylanbrc 572 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
19 | 12, 13, 18 | syl2anb 585 | . . . 4 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
20 | 19 | 3adant1 1124 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
21 | eldifsn 4454 | . . . . 5 ⊢ (𝑧 ∈ (ℝ* ∖ {-∞}) ↔ (𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞)) | |
22 | xaddass 12283 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞) ∧ (𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞)) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) | |
23 | 12, 13, 21, 22 | syl3anb 1164 | . . . 4 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞})) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
24 | 23 | adantl 467 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞}))) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
25 | 0re 10245 | . . . 4 ⊢ 0 ∈ ℝ | |
26 | rexr 10290 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
27 | renemnf 10293 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
28 | eldifsn 4454 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
29 | 26, 27, 28 | sylanbrc 572 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
30 | 25, 29 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
31 | eldifi 3883 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
32 | 31 | adantl 467 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
33 | xaddid2 12277 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
35 | xaddid1 12276 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (𝑥 +𝑒 0) = 𝑥) | |
36 | 32, 35 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
37 | 5, 11, 20, 24, 30, 34, 36 | ismndd 17520 | . 2 ⊢ (⊤ → 𝑅 ∈ Mnd) |
38 | 37 | trud 1641 | 1 ⊢ 𝑅 ∈ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 ∧ w3a 1071 = wceq 1631 ⊤wtru 1632 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∖ cdif 3720 ⊆ wss 3723 {csn 4317 ‘cfv 6030 (class class class)co 6795 ℝcr 10140 0cc0 10141 -∞cmnf 10277 ℝ*cxr 10278 +𝑒 cxad 12148 Basecbs 16063 ↾s cress 16064 +gcplusg 16148 ℝ*𝑠cxrs 16367 Mndcmnd 17501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-om 7216 df-1st 7318 df-2nd 7319 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-oadd 7720 df-er 7899 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-z 11584 df-dec 11700 df-uz 11893 df-xadd 12151 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-tset 16167 df-ple 16168 df-ds 16171 df-xrs 16369 df-mgm 17449 df-sgrp 17491 df-mnd 17502 |
This theorem is referenced by: xrs1cmn 20000 xrge0subm 20001 xrge00 30025 |
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