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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 24817 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | nrgtrg 24727 | . . 3 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ TopRing) | |
3 | eqid 2735 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
4 | 3 | trgtmd 24189 | . . 3 ⊢ (ℂfld ∈ TopRing → (mulGrp‘ℂfld) ∈ TopMnd) |
5 | 1, 2, 4 | mp2b 10 | . 2 ⊢ (mulGrp‘ℂfld) ∈ TopMnd |
6 | unitsscn 13537 | . . 3 ⊢ (0[,]1) ⊆ ℂ | |
7 | 1elunit 13507 | . . 3 ⊢ 1 ∈ (0[,]1) | |
8 | iimulcl 24980 | . . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (𝑥 · 𝑦) ∈ (0[,]1)) | |
9 | 8 | rgen2 3197 | . . 3 ⊢ ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(𝑥 · 𝑦) ∈ (0[,]1) |
10 | nrgring 24700 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
11 | 3 | ringmgp 20257 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
12 | 1, 10, 11 | mp2b 10 | . . . 4 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
13 | cnfldbas 21386 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 3, 13 | mgpbas 20158 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
15 | cnfld1 21424 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
16 | 3, 15 | ringidval 20201 | . . . . 5 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
17 | cnfldmul 21390 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
18 | 3, 17 | mgpplusg 20156 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
19 | 14, 16, 18 | issubm 18829 | . . . 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 24128 | . 2 ⊢ (((mulGrp‘ℂfld) ∈ TopMnd ∧ (0[,]1) ∈ (SubMnd‘(mulGrp‘ℂfld))) → 𝐼 ∈ TopMnd) |
24 | 5, 21, 23 | mp2an 692 | 1 ⊢ 𝐼 ∈ TopMnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 [,]cicc 13387 ↾s cress 17274 Mndcmnd 18760 SubMndcsubmnd 18808 mulGrpcmgp 20152 Ringcrg 20251 ℂfldccnfld 21382 TopMndctmd 24094 TopRingctrg 24180 NrmRingcnrg 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-tmd 24096 df-tgp 24097 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 |
This theorem is referenced by: xrge0iifmhm 33900 xrge0pluscn 33901 xrge0tmd 33906 |
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