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Mirrors > Home > MPE Home > Th. List > xrsmcmn | Structured version Visualization version GIF version |
Description: The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrsmgmdifsgrp 20005.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn | ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2760 | . . . . 5 ⊢ (mulGrp‘ℝ*𝑠) = (mulGrp‘ℝ*𝑠) | |
2 | xrsbas 19984 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | 1, 2 | mgpbas 18715 | . . . 4 ⊢ ℝ* = (Base‘(mulGrp‘ℝ*𝑠)) |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ* = (Base‘(mulGrp‘ℝ*𝑠))) |
5 | xrsmul 19986 | . . . . 5 ⊢ ·e = (.r‘ℝ*𝑠) | |
6 | 1, 5 | mgpplusg 18713 | . . . 4 ⊢ ·e = (+g‘(mulGrp‘ℝ*𝑠)) |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ·e = (+g‘(mulGrp‘ℝ*𝑠))) |
8 | xmulcl 12316 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) | |
9 | 8 | 3adant1 1125 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) |
10 | xmulass 12330 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) | |
11 | 10 | adantl 473 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) |
12 | 1re 10251 | . . . . 5 ⊢ 1 ∈ ℝ | |
13 | rexr 10297 | . . . . 5 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ*) |
15 | xmulid2 12323 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (1 ·e 𝑥) = 𝑥) | |
16 | 15 | adantl 473 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (1 ·e 𝑥) = 𝑥) |
17 | xmulid1 12322 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 1) = 𝑥) | |
18 | 17 | adantl 473 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (𝑥 ·e 1) = 𝑥) |
19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 17534 | . . 3 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ Mnd) |
20 | xmulcom 12309 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) | |
21 | 20 | 3adant1 1125 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) |
22 | 4, 7, 19, 21 | iscmnd 18425 | . 2 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ CMnd) |
23 | 22 | trud 1642 | 1 ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1072 = wceq 1632 ⊤wtru 1633 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6814 ℝcr 10147 1c1 10149 ℝ*cxr 10285 ·e cxmu 12158 Basecbs 16079 +gcplusg 16163 ℝ*𝑠cxrs 16382 CMndccmn 18413 mulGrpcmgp 18709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-xneg 12159 df-xmul 12161 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-plusg 16176 df-mulr 16177 df-tset 16182 df-ple 16183 df-ds 16186 df-xrs 16384 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-cmn 18415 df-mgp 18710 |
This theorem is referenced by: (None) |
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