nvlinv - Metamath Proof Explorer

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Theorem nvlinv 31119

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 31084	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvrinv.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 31071	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2762	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		eqid 2762	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
7	4, 5, 6	grpolinv 30993	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	2, 7	sylan 592	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
9		nvrinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	3, 1, 9, 6	nvinv 31106	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴))
11	10	oveq1d 7431	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (((inv‘𝐺)‘𝐴)𝐺𝐴))
12		nvrinv.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	1, 12	0vfval 31073	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
14	13	adantr 486	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺))
15	8, 11, 14	3eqtr4d 2807	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7416 1c1 11128 -cneg 11469 GrpOpcgr 30956 GIdcgi 30957 invcgn 30958 NrmCVeccnv 31051 +_𝑣 cpv 31052 BaseSetcba 31053 ·_𝑠OLD cns 31054 0_veccn0v 31055

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7989 df-2nd 7990 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-ltxr 11275 df-sub 11470 df-neg 11471 df-grpo 30960 df-gid 30961 df-ginv 30962 df-ablo 31012 df-vc 31026 df-nv 31059 df-va 31062 df-ba 31063 df-sm 31064 df-0v 31065 df-nmcv 31067

This theorem is used by: nvabs 31139 imsmetlem 31157 lno0 31223

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