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Theorem nvlinv 29014

Description: Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvrinv.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvrinv.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvrinv.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nvrinv.6	⊢ 𝑍 = (0_vec‘𝑈)

**Assertion**
Ref	Expression
nvlinv	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

**Proof of Theorem nvlinv**
Step	Hyp	Ref	Expression
1		nvrinv.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2	1	nvgrp 28979	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvrinv.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28966	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2738	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		eqid 2738	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
7	4, 5, 6	grpolinv 28888	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	2, 7	sylan 580	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
9		nvrinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	3, 1, 9, 6	nvinv 29001	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴))
11	10	oveq1d 7290	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (((inv‘𝐺)‘𝐴)𝐺𝐴))
12		nvrinv.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	1, 12	0vfval 28968	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
14	13	adantr 481	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺))
15	8, 11, 14	3eqtr4d 2788	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 1c1 10872 -cneg 11206 GrpOpcgr 28851 GIdcgi 28852 invcgn 28853 NrmCVeccnv 28946 +_𝑣 cpv 28947 BaseSetcba 28948 ·_𝑠OLD cns 28949 0_veccn0v 28950

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-sub 11207 df-neg 11208 df-grpo 28855 df-gid 28856 df-ginv 28857 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-nmcv 28962

This theorem is referenced by: nvabs 29034 imsmetlem 29052 lno0 29118

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