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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 is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 20105.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn | ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . . 5 ⊢ (mulGrp‘ℝ*𝑠) = (mulGrp‘ℝ*𝑠) | |
2 | xrsbas 20084 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | 1, 2 | mgpbas 18811 | . . . 4 ⊢ ℝ* = (Base‘(mulGrp‘ℝ*𝑠)) |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ* = (Base‘(mulGrp‘ℝ*𝑠))) |
5 | xrsmul 20086 | . . . . 5 ⊢ ·e = (.r‘ℝ*𝑠) | |
6 | 1, 5 | mgpplusg 18809 | . . . 4 ⊢ ·e = (+g‘(mulGrp‘ℝ*𝑠)) |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ·e = (+g‘(mulGrp‘ℝ*𝑠))) |
8 | xmulcl 12352 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) | |
9 | 8 | 3adant1 1161 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) |
10 | xmulass 12366 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) | |
11 | 10 | adantl 474 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) |
12 | 1re 10328 | . . . . 5 ⊢ 1 ∈ ℝ | |
13 | rexr 10374 | . . . . 5 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ*) |
15 | xmulid2 12359 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (1 ·e 𝑥) = 𝑥) | |
16 | 15 | adantl 474 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (1 ·e 𝑥) = 𝑥) |
17 | xmulid1 12358 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 1) = 𝑥) | |
18 | 17 | adantl 474 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (𝑥 ·e 1) = 𝑥) |
19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 17628 | . . 3 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ Mnd) |
20 | xmulcom 12345 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) | |
21 | 20 | 3adant1 1161 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) |
22 | 4, 7, 19, 21 | iscmnd 18520 | . 2 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ CMnd) |
23 | 22 | mptru 1661 | 1 ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1108 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 1c1 10225 ℝ*cxr 10362 ·e cxmu 12192 Basecbs 16184 +gcplusg 16267 ℝ*𝑠cxrs 16475 CMndccmn 18508 mulGrpcmgp 18805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-xneg 12193 df-xmul 12195 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-mulr 16281 df-tset 16286 df-ple 16287 df-ds 16289 df-xrs 16477 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-cmn 18510 df-mgp 18806 |
This theorem is referenced by: (None) |
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