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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 2824 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28395 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | nvinv.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28383 | . . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
5 | nvinv.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28385 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvinv.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28384 | . . 3 ⊢ 𝑋 = ran 𝐺 |
9 | nvinv.5 | . . 3 ⊢ 𝑀 = (inv‘𝐺) | |
10 | 4, 6, 8, 9 | vcm 28356 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) |
11 | 2, 10 | sylan 582 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 1c1 10541 -cneg 10874 invcgn 28271 CVecOLDcvc 28338 NrmCVeccnv 28364 +𝑣 cpv 28365 BaseSetcba 28366 ·𝑠OLD cns 28367 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-grpo 28273 df-gid 28274 df-ginv 28275 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-nmcv 28380 |
This theorem is referenced by: nvinvfval 28420 nvmval 28422 nvmfval 28424 nvnegneg 28429 nvrinv 28431 nvlinv 28432 |
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