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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 28652 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvrinv.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28639 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2736 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
6 | eqid 2736 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
7 | 4, 5, 6 | grpolinv 28561 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺)) |
8 | 2, 7 | sylan 583 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺)) |
9 | nvrinv.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 3, 1, 9, 6 | nvinv 28674 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴)) |
11 | 10 | oveq1d 7206 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (((inv‘𝐺)‘𝐴)𝐺𝐴)) |
12 | nvrinv.6 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 1, 12 | 0vfval 28641 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺)) |
14 | 13 | adantr 484 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺)) |
15 | 8, 11, 14 | 3eqtr4d 2781 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 1c1 10695 -cneg 11028 GrpOpcgr 28524 GIdcgi 28525 invcgn 28526 NrmCVeccnv 28619 +𝑣 cpv 28620 BaseSetcba 28621 ·𝑠OLD cns 28622 0veccn0v 28623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-grpo 28528 df-gid 28529 df-ginv 28530 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-nmcv 28635 |
This theorem is referenced by: nvabs 28707 imsmetlem 28725 lno0 28791 |
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