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Mirrors > Home > MPE Home > Th. List > xrs1cmn | Structured version Visualization version GIF version |
Description: The extended real numbers restricted to ℝ* ∖ {-∞} form a commutative monoid. They are not a group because 1 + +∞ = 2 + +∞ even though 1 ≠ 2. (Contributed by Mario Carneiro, 27-Nov-2014.) |
Ref | Expression |
---|---|
xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
Ref | Expression |
---|---|
xrs1cmn | ⊢ 𝑅 ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrs1mnd.1 | . . 3 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1mnd 20858 | . 2 ⊢ 𝑅 ∈ Mnd |
3 | eldifi 4090 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
4 | eldifi 4090 | . . . 4 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) → 𝑦 ∈ ℝ*) | |
5 | xaddcom 13168 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥)) | |
6 | 3, 4, 5 | syl2an 597 | . . 3 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥)) |
7 | 6 | rgen2 3191 | . 2 ⊢ ∀𝑥 ∈ (ℝ* ∖ {-∞})∀𝑦 ∈ (ℝ* ∖ {-∞})(𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥) |
8 | difss 4095 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
9 | xrsbas 20836 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
10 | 1, 9 | ressbas2 17128 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
12 | xrex 12920 | . . . . 5 ⊢ ℝ* ∈ V | |
13 | 12 | difexi 5289 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
14 | xrsadd 20837 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
15 | 1, 14 | ressplusg 17179 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
17 | 11, 16 | iscmn 19579 | . 2 ⊢ (𝑅 ∈ CMnd ↔ (𝑅 ∈ Mnd ∧ ∀𝑥 ∈ (ℝ* ∖ {-∞})∀𝑦 ∈ (ℝ* ∖ {-∞})(𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥))) |
18 | 2, 7, 17 | mpbir2an 710 | 1 ⊢ 𝑅 ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 {csn 4590 ‘cfv 6500 (class class class)co 7361 -∞cmnf 11195 ℝ*cxr 11196 +𝑒 cxad 13039 Basecbs 17091 ↾s cress 17120 +gcplusg 17141 ℝ*𝑠cxrs 17390 Mndcmnd 18564 CMndccmn 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-xadd 13042 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-tset 17160 df-ple 17161 df-ds 17163 df-xrs 17392 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-cmn 19572 |
This theorem is referenced by: xrge0cmn 20862 imasdsf1olem 23749 gsumge0cl 44702 |
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