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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 2818 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | nvinvfval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
3 | 1, 2 | nvsf 28323 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
4 | neg1cn 11739 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | nvinvfval.3 | . . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) | |
6 | 5 | curry1f 7790 | . . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
7 | 3, 4, 6 | sylancl 586 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
8 | 7 | ffnd 6508 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈)) |
9 | nvinvfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
10 | 9 | nvgrp 28321 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
11 | 1, 9 | bafval 28308 | . . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺 |
12 | eqid 2818 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
13 | 11, 12 | grpoinvf 28236 | . . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈)) |
14 | f1ofn 6609 | . . 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 6508 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
18 | 5 | curry1val 7789 | . . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
19 | 17, 4, 18 | sylancl 586 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
20 | 1, 9, 2, 12 | nvinv 28343 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥)) |
21 | 19, 20 | eqtrd 2853 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥)) |
22 | 8, 15, 21 | eqfnfvd 6797 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 × cxp 5546 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 2nd c2nd 7677 ℂcc 10523 1c1 10526 -cneg 10859 GrpOpcgr 28193 invcgn 28195 NrmCVeccnv 28288 +𝑣 cpv 28289 BaseSetcba 28290 ·𝑠OLD cns 28291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 df-grpo 28197 df-gid 28198 df-ginv 28199 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 |
This theorem is referenced by: hhssabloilem 28965 |
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