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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 2622 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | nvinvfval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
3 | 1, 2 | nvsf 27474 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
4 | neg1cn 11124 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | nvinvfval.3 | . . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) | |
6 | 5 | curry1f 7271 | . . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
7 | 3, 4, 6 | sylancl 694 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
8 | ffn 6045 | . . 3 ⊢ (𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈) → 𝑁 Fn (BaseSet‘𝑈)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈)) |
10 | nvinvfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
11 | 10 | nvgrp 27472 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
12 | 1, 10 | bafval 27459 | . . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺 |
13 | eqid 2622 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
14 | 12, 13 | grpoinvf 27386 | . . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈)) |
15 | f1ofn 6138 | . . 3 ⊢ ((inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈) → (inv‘𝐺) Fn (BaseSet‘𝑈)) | |
16 | 11, 14, 15 | 3syl 18 | . 2 ⊢ (𝑈 ∈ NrmCVec → (inv‘𝐺) Fn (BaseSet‘𝑈)) |
17 | ffn 6045 | . . . . . 6 ⊢ (𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) | |
18 | 3, 17 | syl 17 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
20 | 5 | curry1val 7270 | . . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
21 | 19, 4, 20 | sylancl 694 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
22 | 1, 10, 2, 13 | nvinv 27494 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥)) |
23 | 21, 22 | eqtrd 2656 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥)) |
24 | 9, 16, 23 | eqfnfvd 6314 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 2nd c2nd 7167 ℂcc 9934 1c1 9937 -cneg 10267 GrpOpcgr 27343 invcgn 27345 NrmCVeccnv 27439 +𝑣 cpv 27440 BaseSetcba 27441 ·𝑠OLD cns 27442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 |
This theorem is referenced by: hhssabloilem 28118 |
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