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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 20585 | . 2 ⊢ 𝑅 ∈ Mnd |
3 | eldifi 4105 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
4 | eldifi 4105 | . . . 4 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) → 𝑦 ∈ ℝ*) | |
5 | xaddcom 12636 | . . . 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 3205 | . 2 ⊢ ∀𝑥 ∈ (ℝ* ∖ {-∞})∀𝑦 ∈ (ℝ* ∖ {-∞})(𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥) |
8 | difss 4110 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
9 | xrsbas 20563 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
10 | 1, 9 | ressbas2 16557 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
12 | xrex 12389 | . . . . 5 ⊢ ℝ* ∈ V | |
13 | 12 | difexi 5234 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
14 | xrsadd 20564 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
15 | 1, 14 | ressplusg 16614 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
17 | 11, 16 | iscmn 18916 | . 2 ⊢ (𝑅 ∈ CMnd ↔ (𝑅 ∈ Mnd ∧ ∀𝑥 ∈ (ℝ* ∖ {-∞})∀𝑦 ∈ (ℝ* ∖ {-∞})(𝑥 +𝑒 𝑦) = (𝑦 +𝑒 𝑥))) |
18 | 2, 7, 17 | mpbir2an 709 | 1 ⊢ 𝑅 ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 -∞cmnf 10675 ℝ*cxr 10676 +𝑒 cxad 12508 Basecbs 16485 ↾s cress 16486 +gcplusg 16567 ℝ*𝑠cxrs 16775 Mndcmnd 17913 CMndccmn 18908 |
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-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-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-xrs 16777 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-cmn 18910 |
This theorem is referenced by: xrge0cmn 20589 imasdsf1olem 22985 gsumge0cl 42660 |
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