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Mirrors > Home > MPE Home > Th. List > nvinvfval | Structured version Visualization version GIF version |
Description: Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvinvfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvinvfval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvinvfval.3 | ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) |
Ref | Expression |
---|---|
nvinvfval | ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | nvinvfval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
3 | 1, 2 | nvsf 27999 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
4 | neg1cn 11434 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | nvinvfval.3 | . . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) | |
6 | 5 | curry1f 7508 | . . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
7 | 3, 4, 6 | sylancl 581 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
8 | 7 | ffnd 6257 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈)) |
9 | nvinvfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
10 | 9 | nvgrp 27997 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
11 | 1, 9 | bafval 27984 | . . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺 |
12 | eqid 2799 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
13 | 11, 12 | grpoinvf 27912 | . . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈)) |
14 | f1ofn 6357 | . . 3 ⊢ ((inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈) → (inv‘𝐺) Fn (BaseSet‘𝑈)) | |
15 | 10, 13, 14 | 3syl 18 | . 2 ⊢ (𝑈 ∈ NrmCVec → (inv‘𝐺) Fn (BaseSet‘𝑈)) |
16 | 3 | ffnd 6257 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
17 | 16 | adantr 473 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
18 | 5 | curry1val 7507 | . . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
19 | 17, 4, 18 | sylancl 581 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
20 | 1, 9, 2, 12 | nvinv 28019 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥)) |
21 | 19, 20 | eqtrd 2833 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥)) |
22 | 8, 15, 21 | eqfnfvd 6540 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 {csn 4368 × cxp 5310 ◡ccnv 5311 ↾ cres 5314 ∘ ccom 5316 Fn wfn 6096 ⟶wf 6097 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 2nd c2nd 7400 ℂcc 10222 1c1 10225 -cneg 10557 GrpOpcgr 27869 invcgn 27871 NrmCVeccnv 27964 +𝑣 cpv 27965 BaseSetcba 27966 ·𝑠OLD cns 27967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-neg 10559 df-grpo 27873 df-gid 27874 df-ginv 27875 df-ablo 27925 df-vc 27939 df-nv 27972 df-va 27975 df-ba 27976 df-sm 27977 df-0v 27978 df-nmcv 27980 |
This theorem is referenced by: hhssabloilem 28643 |
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