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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 25178 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
3 | cvsdiv.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | clm0 25124 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘𝐹)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑊 ∈ ℂVec → 0 = (0g‘𝐹)) |
6 | 5 | sneqd 4660 | . . 3 ⊢ (𝑊 ∈ ℂVec → {0} = {(0g‘𝐹)}) |
7 | 6 | difeq2d 4149 | . 2 ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (𝐾 ∖ {(0g‘𝐹)})) |
8 | 1 | cvslvec 25177 | . . 3 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ LVec) |
9 | 3 | lvecdrng 21127 | . . 3 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
10 | cvsdiv.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
11 | eqid 2740 | . . . . 5 ⊢ (Unit‘𝐹) = (Unit‘𝐹) | |
12 | eqid 2740 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
13 | 10, 11, 12 | isdrng 20755 | . . . 4 ⊢ (𝐹 ∈ DivRing ↔ (𝐹 ∈ Ring ∧ (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)}))) |
14 | 13 | simprbi 496 | . . 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 2783 | 1 ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 {csn 4648 ‘cfv 6573 0cc0 11184 Basecbs 17258 Scalarcsca 17314 0gc0g 17499 Ringcrg 20260 Unitcui 20381 DivRingcdr 20751 LVecclvec 21124 ℂModcclm 25114 ℂVecccvs 25175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-addf 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-subg 19163 df-cmn 19824 df-mgp 20162 df-ring 20262 df-cring 20263 df-subrg 20597 df-drng 20753 df-lvec 21125 df-cnfld 21388 df-clm 25115 df-cvs 25176 |
This theorem is referenced by: cvsdiv 25184 cvsdivcl 25185 |
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