nvinv - Metamath Proof Explorer

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Theorem nvinv 28902

Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvinv.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvinv.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvinv.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nvinv.5	⊢ 𝑀 = (inv‘𝐺)

**Assertion**
Ref	Expression
nvinv	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

**Proof of Theorem nvinv**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28878	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		nvinv.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	3	vafval 28866	. . 3 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
5		nvinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28868	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvinv.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 28867	. . 3 ⊢ 𝑋 = ran 𝐺
9		nvinv.5	. . 3 ⊢ 𝑀 = (inv‘𝐺)
10	4, 6, 8, 9	vcm 28839	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))
11	2, 10	sylan 579	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 1c1 10803 -cneg 11136 invcgn 28754 CVec_OLDcvc 28821 NrmCVeccnv 28847 +_𝑣 cpv 28848 BaseSetcba 28849 ·_𝑠OLD cns 28850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-ltxr 10945 df-sub 11137 df-neg 11138 df-grpo 28756 df-gid 28757 df-ginv 28758 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-nmcv 28863

This theorem is referenced by: nvinvfval 28903 nvmval 28905 nvmfval 28907 nvnegneg 28912 nvrinv 28914 nvlinv 28915

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