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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 21353.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn | ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . 5 ⊢ (mulGrp‘ℝ*𝑠) = (mulGrp‘ℝ*𝑠) | |
2 | xrsbas 21328 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | 1, 2 | mgpbas 20092 | . . . 4 ⊢ ℝ* = (Base‘(mulGrp‘ℝ*𝑠)) |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ* = (Base‘(mulGrp‘ℝ*𝑠))) |
5 | xrsmul 21330 | . . . . 5 ⊢ ·e = (.r‘ℝ*𝑠) | |
6 | 1, 5 | mgpplusg 20090 | . . . 4 ⊢ ·e = (+g‘(mulGrp‘ℝ*𝑠)) |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ·e = (+g‘(mulGrp‘ℝ*𝑠))) |
8 | xmulcl 13287 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) | |
9 | 8 | 3adant1 1127 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) |
10 | xmulass 13301 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) | |
11 | 10 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) |
12 | 1re 11246 | . . . . 5 ⊢ 1 ∈ ℝ | |
13 | rexr 11292 | . . . . 5 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ*) |
15 | xmullid 13294 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (1 ·e 𝑥) = 𝑥) | |
16 | 15 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (1 ·e 𝑥) = 𝑥) |
17 | xmulrid 13293 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 1) = 𝑥) | |
18 | 17 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (𝑥 ·e 1) = 𝑥) |
19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 18719 | . . 3 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ Mnd) |
20 | xmulcom 13280 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) | |
21 | 20 | 3adant1 1127 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) |
22 | 4, 7, 19, 21 | iscmnd 19761 | . 2 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ CMnd) |
23 | 22 | mptru 1540 | 1 ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 1c1 11141 ℝ*cxr 11279 ·e cxmu 13126 Basecbs 17183 +gcplusg 17236 ℝ*𝑠cxrs 17485 CMndccmn 19747 mulGrpcmgp 20086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-xneg 13127 df-xmul 13129 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-tset 17255 df-ple 17256 df-ds 17258 df-xrs 17487 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-cmn 19749 df-mgp 20087 |
This theorem is referenced by: (None) |
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