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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 2726 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | nvinvfval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
3 | 1, 2 | nvsf 30552 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
4 | neg1cn 12378 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | nvinvfval.3 | . . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) | |
6 | 5 | curry1f 8120 | . . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
7 | 3, 4, 6 | sylancl 584 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈)) |
8 | 7 | ffnd 6729 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈)) |
9 | nvinvfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
10 | 9 | nvgrp 30550 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
11 | 1, 9 | bafval 30537 | . . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺 |
12 | eqid 2726 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
13 | 11, 12 | grpoinvf 30465 | . . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈)) |
14 | f1ofn 6844 | . . 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 6729 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
17 | 16 | adantr 479 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
18 | 5 | curry1val 8119 | . . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
19 | 17, 4, 18 | sylancl 584 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥)) |
20 | 1, 9, 2, 12 | nvinv 30572 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥)) |
21 | 19, 20 | eqtrd 2766 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥)) |
22 | 8, 15, 21 | eqfnfvd 7047 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4633 × cxp 5680 ◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6549 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 2nd c2nd 8002 ℂcc 11156 1c1 11159 -cneg 11495 GrpOpcgr 30422 invcgn 30424 NrmCVeccnv 30517 +𝑣 cpv 30518 BaseSetcba 30519 ·𝑠OLD cns 30520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-neg 11497 df-grpo 30426 df-gid 30427 df-ginv 30428 df-ablo 30478 df-vc 30492 df-nv 30525 df-va 30528 df-ba 30529 df-sm 30530 df-0v 30531 df-nmcv 30533 |
This theorem is referenced by: hhssabloilem 31194 |
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