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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 19602.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn | ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . . 5 ⊢ (mulGrp‘ℝ*𝑠) = (mulGrp‘ℝ*𝑠) | |
2 | xrsbas 19581 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | 1, 2 | mgpbas 18318 | . . . 4 ⊢ ℝ* = (Base‘(mulGrp‘ℝ*𝑠)) |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ* = (Base‘(mulGrp‘ℝ*𝑠))) |
5 | xrsmul 19583 | . . . . 5 ⊢ ·e = (.r‘ℝ*𝑠) | |
6 | 1, 5 | mgpplusg 18316 | . . . 4 ⊢ ·e = (+g‘(mulGrp‘ℝ*𝑠)) |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ·e = (+g‘(mulGrp‘ℝ*𝑠))) |
8 | xmulcl 11975 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) | |
9 | 8 | 3adant1 1072 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*) |
10 | xmulass 11989 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) | |
11 | 10 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))) |
12 | 1re 9918 | . . . . 5 ⊢ 1 ∈ ℝ | |
13 | rexr 9964 | . . . . 5 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ*) |
15 | xmulid2 11982 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (1 ·e 𝑥) = 𝑥) | |
16 | 15 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (1 ·e 𝑥) = 𝑥) |
17 | xmulid1 11981 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 1) = 𝑥) | |
18 | 17 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → (𝑥 ·e 1) = 𝑥) |
19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 17136 | . . 3 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ Mnd) |
20 | xmulcom 11968 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) | |
21 | 20 | 3adant1 1072 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) = (𝑦 ·e 𝑥)) |
22 | 4, 7, 19, 21 | iscmnd 18028 | . 2 ⊢ (⊤ → (mulGrp‘ℝ*𝑠) ∈ CMnd) |
23 | 22 | trud 1484 | 1 ⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 ℝ*cxr 9952 ·e cxmu 11821 Basecbs 15695 +gcplusg 15768 ℝ*𝑠cxrs 15983 CMndccmn 18016 mulGrpcmgp 18312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-xneg 11822 df-xmul 11824 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-tset 15787 df-ple 15788 df-ds 15791 df-xrs 15985 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-mgp 18313 |
This theorem is referenced by: (None) |
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