Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iistmd | Structured version Visualization version GIF version |
Description: The closed unit forms a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
Ref | Expression |
---|---|
df-iis | ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) |
Ref | Expression |
---|---|
iistmd | ⊢ 𝐼 ∈ TopMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnrg 22584 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | nrgtrg 22494 | . . 3 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ TopRing) | |
3 | eqid 2622 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
4 | 3 | trgtmd 21968 | . . 3 ⊢ (ℂfld ∈ TopRing → (mulGrp‘ℂfld) ∈ TopMnd) |
5 | 1, 2, 4 | mp2b 10 | . 2 ⊢ (mulGrp‘ℂfld) ∈ TopMnd |
6 | unitsscn 29942 | . . 3 ⊢ (0[,]1) ⊆ ℂ | |
7 | 1elunit 12291 | . . 3 ⊢ 1 ∈ (0[,]1) | |
8 | iimulcl 22736 | . . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (𝑥 · 𝑦) ∈ (0[,]1)) | |
9 | 8 | rgen2a 2977 | . . 3 ⊢ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1) |
10 | nrgring 22467 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
11 | 3 | ringmgp 18553 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
12 | 1, 10, 11 | mp2b 10 | . . . 4 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
13 | cnfldbas 19750 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 3, 13 | mgpbas 18495 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
15 | cnfld1 19771 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
16 | 3, 15 | ringidval 18503 | . . . . 5 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
17 | cnfldmul 19752 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
18 | 3, 17 | mgpplusg 18493 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
19 | 14, 16, 18 | issubm 17347 | . . . 4 ⊢ ((mulGrp‘ℂfld) ∈ Mnd → ((0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld)) ↔ ((0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1) ∧ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1)))) |
20 | 12, 19 | ax-mp 5 | . . 3 ⊢ ((0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld)) ↔ ((0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1) ∧ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1))) |
21 | 6, 7, 9, 20 | mpbir3an 1244 | . 2 ⊢ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
22 | df-iis | . . 3 ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
23 | 22 | submtmd 21908 | . 2 ⊢ (((mulGrp‘ℂfld) ∈ TopMnd ∧ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld))) → 𝐼 ∈ TopMnd) |
24 | 5, 21, 23 | mp2an 708 | 1 ⊢ 𝐼 ∈ TopMnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 [,]cicc 12178 ↾s cress 15858 Mndcmnd 17294 SubMndcsubmnd 17334 mulGrpcmgp 18489 Ringcrg 18547 ℂfldccnfld 19746 TopMndctmd 21874 TopRingctrg 21959 NrmRingcnrg 22384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-tmd 21876 df-tgp 21877 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 |
This theorem is referenced by: xrge0iifmhm 29985 xrge0pluscn 29986 xrge0tmd 29992 |
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