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Mirrors > Home > MPE Home > Th. List > cvsunit | Structured version Visualization version GIF version |
Description: Unit group of the scalar ring of a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvsdiv.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cvsdiv.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cvsunit | ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
2 | 1 | cvsclm 23118 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
3 | cvsdiv.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | clm0 23064 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘𝐹)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑊 ∈ ℂVec → 0 = (0g‘𝐹)) |
6 | 5 | sneqd 4325 | . . 3 ⊢ (𝑊 ∈ ℂVec → {0} = {(0g‘𝐹)}) |
7 | 6 | difeq2d 3863 | . 2 ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (𝐾 ∖ {(0g‘𝐹)})) |
8 | 1 | cvslvec 23117 | . . 3 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ LVec) |
9 | 3 | lvecdrng 19299 | . . 3 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
10 | cvsdiv.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
11 | eqid 2752 | . . . . 5 ⊢ (Unit‘𝐹) = (Unit‘𝐹) | |
12 | eqid 2752 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
13 | 10, 11, 12 | isdrng 18945 | . . . 4 ⊢ (𝐹 ∈ DivRing ↔ (𝐹 ∈ Ring ∧ (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)}))) |
14 | 13 | simprbi 483 | . . 3 ⊢ (𝐹 ∈ DivRing → (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)})) |
15 | 8, 9, 14 | 3syl 18 | . 2 ⊢ (𝑊 ∈ ℂVec → (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)})) |
16 | 7, 15 | eqtr4d 2789 | 1 ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ∖ cdif 3704 {csn 4313 ‘cfv 6041 0cc0 10120 Basecbs 16051 Scalarcsca 16138 0gc0g 16294 Ringcrg 18739 Unitcui 18831 DivRingcdr 18941 LVecclvec 19296 ℂModcclm 23054 ℂVecccvs 23115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-addf 10199 ax-mulf 10200 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-fz 12512 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-0g 16296 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-grp 17618 df-subg 17784 df-cmn 18387 df-mgp 18682 df-ring 18741 df-cring 18742 df-drng 18943 df-subrg 18972 df-lvec 19297 df-cnfld 19941 df-clm 23055 df-cvs 23116 |
This theorem is referenced by: cvsdiv 23124 cvsdivcl 23125 |
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