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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 24648 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | nrgtrg 24558 | . . 3 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ TopRing) | |
3 | eqid 2726 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
4 | 3 | trgtmd 24020 | . . 3 ⊢ (ℂfld ∈ TopRing → (mulGrp‘ℂfld) ∈ TopMnd) |
5 | 1, 2, 4 | mp2b 10 | . 2 ⊢ (mulGrp‘ℂfld) ∈ TopMnd |
6 | unitsscn 13480 | . . 3 ⊢ (0[,]1) ⊆ ℂ | |
7 | 1elunit 13450 | . . 3 ⊢ 1 ∈ (0[,]1) | |
8 | iimulcl 24811 | . . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (𝑥 · 𝑦) ∈ (0[,]1)) | |
9 | 8 | rgen2 3191 | . . 3 ⊢ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1) |
10 | nrgring 24531 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
11 | 3 | ringmgp 20142 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
12 | 1, 10, 11 | mp2b 10 | . . . 4 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
13 | cnfldbas 21240 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 3, 13 | mgpbas 20043 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
15 | cnfld1 21278 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
16 | 3, 15 | ringidval 20086 | . . . . 5 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
17 | cnfldmul 21244 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
18 | 3, 17 | mgpplusg 20041 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
19 | 14, 16, 18 | issubm 18726 | . . . 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 1338 | . 2 ⊢ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
22 | df-iis | . . 3 ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
23 | 22 | submtmd 23959 | . 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 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 0cc0 11109 1c1 11110 · cmul 11114 [,]cicc 13330 ↾s cress 17180 Mndcmnd 18665 SubMndcsubmnd 18710 mulGrpcmgp 20037 Ringcrg 20136 ℂfldccnfld 21236 TopMndctmd 23925 TopRingctrg 24011 NrmRingcnrg 24439 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-plusf 18570 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-subrng 20444 df-subrg 20469 df-abv 20658 df-lmod 20706 df-scaf 20707 df-sra 21019 df-rgmod 21020 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cn 23082 df-cnp 23083 df-tx 23417 df-hmeo 23610 df-tmd 23927 df-tgp 23928 df-trg 24015 df-xms 24177 df-ms 24178 df-tms 24179 df-nm 24442 df-ngp 24443 df-nrg 24445 df-nlm 24446 |
This theorem is referenced by: xrge0iifmhm 33449 xrge0pluscn 33450 xrge0tmd 33455 |
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