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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 20510.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn | ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (mulGrp‘ℝ*𝑠) = (mulGrp‘ℝ*𝑠) | |
2 | xrsbas 20489 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | 1, 2 | mgpbas 19174 | . . . 4 ⊢ ℝ* = (Base‘(mulGrp‘ℝ*𝑠)) |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ* = (Base‘(mulGrp‘ℝ*𝑠))) |
5 | xrsmul 20491 | . . . . 5 ⊢ ·e = (.r‘ℝ*𝑠) | |
6 | 1, 5 | mgpplusg 19172 | . . . 4 ⊢ ·e = (+g‘(mulGrp‘ℝ*𝑠)) |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ·e = (+g‘(mulGrp‘ℝ*𝑠))) |
8 | xmulcl 12654 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) | |
9 | 8 | 3adant1 1122 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) |
10 | xmulass 12668 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) | |
11 | 10 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) |
12 | 1re 10629 | . . . . 5 ⊢ 1 ∈ ℝ | |
13 | rexr 10675 | . . . . 5 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ*) |
15 | xmulid2 12661 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (1 ·e 𝑥) = 𝑥) | |
16 | 15 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (1 ·e 𝑥) = 𝑥) |
17 | xmulid1 12660 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 1) = 𝑥) | |
18 | 17 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (𝑥 ·e 1) = 𝑥) |
19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 17921 | . . 3 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ Mnd) |
20 | xmulcom 12647 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) | |
21 | 20 | 3adant1 1122 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) |
22 | 4, 7, 19, 21 | iscmnd 18848 | . 2 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ CMnd) |
23 | 22 | mptru 1535 | 1 ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1079 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 1c1 10526 ℝ*cxr 10662 ·e cxmu 12494 Basecbs 16471 +gcplusg 16553 ℝ*𝑠cxrs 16761 CMndccmn 18835 mulGrpcmgp 19168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-xneg 12495 df-xmul 12497 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-tset 16572 df-ple 16573 df-ds 16575 df-xrs 16763 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-cmn 18837 df-mgp 19169 |
This theorem is referenced by: (None) |
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