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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 20976 | . 2 ⊢ 𝑅 ∈ Mnd |
3 | eldifi 4126 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
4 | eldifi 4126 | . . . 4 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) → 𝑦 ∈ ℝ*) | |
5 | xaddcom 13216 | . . . 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 3198 | . 2 ⊢ ∀𝑥 ∈ (ℝ* ∖ {-∞})∀𝑦 ∈ (ℝ* ∖ {-∞})(𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥) |
8 | difss 4131 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
9 | xrsbas 20954 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
10 | 1, 9 | ressbas2 17179 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
12 | xrex 12968 | . . . . 5 ⊢ ℝ* ∈ V | |
13 | 12 | difexi 5328 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
14 | xrsadd 20955 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
15 | 1, 14 | ressplusg 17232 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
17 | 11, 16 | iscmn 19652 | . 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 3062 Vcvv 3475 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ‘cfv 6541 (class class class)co 7406 -∞cmnf 11243 ℝ*cxr 11244 +𝑒 cxad 13087 Basecbs 17141 ↾s cress 17170 +gcplusg 17194 ℝ*𝑠cxrs 17443 Mndcmnd 18622 CMndccmn 19643 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-xadd 13090 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-tset 17213 df-ple 17214 df-ds 17216 df-xrs 17445 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-cmn 19645 |
This theorem is referenced by: xrge0cmn 20980 imasdsf1olem 23871 gsumge0cl 45074 |
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