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Theorem nvlinv 28431

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 28396	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvrinv.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28383	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2823	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		eqid 2823	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
7	4, 5, 6	grpolinv 28305	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	2, 7	sylan 582	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((inv‘𝐺)‘𝐴)𝐺𝐴) = (GId‘𝐺))
9		nvrinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	3, 1, 9, 6	nvinv 28418	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘𝐺)‘𝐴))
11	10	oveq1d 7173	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (((inv‘𝐺)‘𝐴)𝐺𝐴))
12		nvrinv.6	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
13	1, 12	0vfval 28385	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
14	13	adantr 483	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘𝐺))
15	8, 11, 14	3eqtr4d 2868	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 1c1 10540 -cneg 10873 GrpOpcgr 28268 GIdcgi 28269 invcgn 28270 NrmCVeccnv 28363 +_𝑣 cpv 28364 BaseSetcba 28365 ·_𝑠OLD cns 28366 0_veccn0v 28367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-grpo 28272 df-gid 28273 df-ginv 28274 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-nmcv 28379

This theorem is referenced by: nvabs 28451 imsmetlem 28469 lno0 28535

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