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Mirrors > Home > MPE Home > Th. List > nvinv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nvinv.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvinv.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvinv.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvinv.5 | ⊢ 𝑀 = (inv‘𝐺) |
Ref | Expression |
---|---|
nvinv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28553 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | nvinv.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28541 | . . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
5 | nvinv.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28543 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvinv.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28542 | . . 3 ⊢ 𝑋 = ran 𝐺 |
9 | nvinv.5 | . . 3 ⊢ 𝑀 = (inv‘𝐺) | |
10 | 4, 6, 8, 9 | vcm 28514 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) |
11 | 2, 10 | sylan 583 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 1st c1st 7715 1c1 10619 -cneg 10952 invcgn 28429 CVecOLDcvc 28496 NrmCVeccnv 28522 +𝑣 cpv 28523 BaseSetcba 28524 ·𝑠OLD cns 28525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-ltxr 10761 df-sub 10953 df-neg 10954 df-grpo 28431 df-gid 28432 df-ginv 28433 df-ablo 28483 df-vc 28497 df-nv 28530 df-va 28533 df-ba 28534 df-sm 28535 df-0v 28536 df-nmcv 28538 |
This theorem is referenced by: nvinvfval 28578 nvmval 28580 nvmfval 28582 nvnegneg 28587 nvrinv 28589 nvlinv 28590 |
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