nvrinv - Metamath Proof Explorer

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Theorem nvrinv 30745

Description: A vector minus 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
nvrinv	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍)

**Proof of Theorem nvrinv**
Step	Hyp	Ref	Expression
1		nvrinv.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2	1	nvgrp 30711	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvrinv.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 30698	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2737	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		eqid 2737	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
7	4, 5, 6	grporinv 30621	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = (GId‘𝐺))
8	2, 7	sylan 581	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = (GId‘𝐺))
9		nvrinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	3, 1, 9, 6	nvinv 30733	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴))
11	10	oveq2d 7386	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
12		nvrinv.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	1, 12	0vfval 30700	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
14	13	adantr 480	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺))
15	8, 11, 14	3eqtr4d 2782	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ‘cfv 6502 (class class class)co 7370 1c1 11041 -cneg 11379 GrpOpcgr 30583 GIdcgi 30584 invcgn 30585 NrmCVeccnv 30678 +_𝑣 cpv 30679 BaseSetcba 30680 ·_𝑠OLD cns 30681 0_veccn0v 30682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-po 5542 df-so 5543 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-1st 7945 df-2nd 7946 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-pnf 11182 df-mnf 11183 df-ltxr 11185 df-sub 11380 df-neg 11381 df-grpo 30587 df-gid 30588 df-ginv 30589 df-ablo 30639 df-vc 30653 df-nv 30686 df-va 30689 df-ba 30690 df-sm 30691 df-0v 30692 df-nmcv 30694

This theorem is referenced by: nvpncan2 30747 ipidsq 30804 ip2i 30922 ipdirilem 30923 ipasslem2 30926

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