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 interval forms a topological monoid under multiplication. (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 23955 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | nrgtrg 23865 | . . 3 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ TopRing) | |
3 | eqid 2740 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
4 | 3 | trgtmd 23327 | . . 3 ⊢ (ℂfld ∈ TopRing → (mulGrp‘ℂfld) ∈ TopMnd) |
5 | 1, 2, 4 | mp2b 10 | . 2 ⊢ (mulGrp‘ℂfld) ∈ TopMnd |
6 | unitsscn 13243 | . . 3 ⊢ (0[,]1) ⊆ ℂ | |
7 | 1elunit 13213 | . . 3 ⊢ 1 ∈ (0[,]1) | |
8 | iimulcl 24111 | . . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (𝑥 · 𝑦) ∈ (0[,]1)) | |
9 | 8 | rgen2 3129 | . . 3 ⊢ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1) |
10 | nrgring 23838 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
11 | 3 | ringmgp 19800 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
12 | 1, 10, 11 | mp2b 10 | . . . 4 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
13 | cnfldbas 20612 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 3, 13 | mgpbas 19737 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
15 | cnfld1 20634 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
16 | 3, 15 | ringidval 19750 | . . . . 5 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
17 | cnfldmul 20614 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
18 | 3, 17 | mgpplusg 19735 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
19 | 14, 16, 18 | issubm 18453 | . . . 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 1340 | . 2 ⊢ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
22 | df-iis | . . 3 ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
23 | 22 | submtmd 23266 | . 2 ⊢ (((mulGrp‘ℂfld) ∈ TopMnd ∧ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld))) → 𝐼 ∈ TopMnd) |
24 | 5, 21, 23 | mp2an 689 | 1 ⊢ 𝐼 ∈ TopMnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ⊆ wss 3892 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 0cc0 10882 1c1 10883 · cmul 10887 [,]cicc 13093 ↾s cress 16952 Mndcmnd 18396 SubMndcsubmnd 18440 mulGrpcmgp 19731 Ringcrg 19794 ℂfldccnfld 20608 TopMndctmd 23232 TopRingctrg 23318 NrmRingcnrg 23746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-map 8609 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-seq 13733 df-exp 13794 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-plusf 18336 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-grp 18591 df-minusg 18592 df-sbg 18593 df-mulg 18712 df-subg 18763 df-cntz 18934 df-cmn 19399 df-abl 19400 df-mgp 19732 df-ur 19749 df-ring 19796 df-cring 19797 df-subrg 20033 df-abv 20088 df-lmod 20136 df-scaf 20137 df-sra 20445 df-rgmod 20446 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cn 22389 df-cnp 22390 df-tx 22724 df-hmeo 22917 df-tmd 23234 df-tgp 23235 df-trg 23322 df-xms 23484 df-ms 23485 df-tms 23486 df-nm 23749 df-ngp 23750 df-nrg 23752 df-nlm 23753 |
This theorem is referenced by: xrge0iifmhm 31898 xrge0pluscn 31899 xrge0tmd 31904 |
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