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Mirrors > Home > MPE Home > Th. List > nvlinv | Structured version Visualization version GIF version |
Description: Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvrinv.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvrinv.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvrinv.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvrinv.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nvlinv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrinv.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvgrp 29848 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvrinv.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 29835 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2733 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
6 | eqid 2733 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
7 | 4, 5, 6 | grpolinv 29757 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺)) |
8 | 2, 7 | sylan 581 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺)) |
9 | nvrinv.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 3, 1, 9, 6 | nvinv 29870 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴)) |
11 | 10 | oveq1d 7419 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (((inv‘𝐺)‘𝐴)𝐺𝐴)) |
12 | nvrinv.6 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 1, 12 | 0vfval 29837 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
14 | 13 | adantr 482 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺)) |
15 | 8, 11, 14 | 3eqtr4d 2783 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 1c1 11107 -cneg 11441 GrpOpcgr 29720 GIdcgi 29721 invcgn 29722 NrmCVeccnv 29815 +𝑣 cpv 29816 BaseSetcba 29817 ·𝑠OLD cns 29818 0veccn0v 29819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 df-grpo 29724 df-gid 29725 df-ginv 29726 df-ablo 29776 df-vc 29790 df-nv 29823 df-va 29826 df-ba 29827 df-sm 29828 df-0v 29829 df-nmcv 29831 |
This theorem is referenced by: nvabs 29903 imsmetlem 29921 lno0 29987 |
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