nvinv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nvinv		Structured version Visualization version GIF version

Theorem nvinv 30568

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 2729	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30544	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		nvinv.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	3	vafval 30532	. . 3 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
5		nvinv.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30534	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvinv.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30533	. . 3 ⊢ 𝑋 = ran 𝐺
9		nvinv.5	. . 3 ⊢ 𝑀 = (inv‘𝐺)
10	4, 6, 8, 9	vcm 30505	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))
11	2, 10	sylan 580	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 1c1 11069 -cneg 11406 invcgn 30420 CVec_OLDcvc 30487 NrmCVeccnv 30513 +_𝑣 cpv 30514 BaseSetcba 30515 ·_𝑠OLD cns 30516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-ltxr 11213 df-sub 11407 df-neg 11408 df-grpo 30422 df-gid 30423 df-ginv 30424 df-ablo 30474 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-nmcv 30529

This theorem is referenced by: nvinvfval 30569 nvmval 30571 nvmfval 30573 nvnegneg 30578 nvrinv 30580 nvlinv 30581

W3C validator