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Mirrors > Home > MPE Home > Th. List > nvnegneg | Structured version Visualization version GIF version |
Description: Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvnegneg.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvnegneg.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvnegneg | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11743 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | nvnegneg.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | nvnegneg.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | 2, 3 | nvscl 28395 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋) |
5 | 1, 4 | mp3an2 1443 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋) |
6 | eqid 2819 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
7 | eqid 2819 | . . . 4 ⊢ (inv‘( +𝑣 ‘𝑈)) = (inv‘( +𝑣 ‘𝑈)) | |
8 | 2, 6, 3, 7 | nvinv 28408 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1𝑆𝐴) ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴))) |
9 | 5, 8 | syldan 593 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴))) |
10 | 2, 6, 3, 7 | nvinv 28408 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘( +𝑣 ‘𝑈))‘𝐴)) |
11 | 10 | fveq2d 6667 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴))) |
12 | 6 | nvgrp 28386 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
13 | 2, 6 | bafval 28373 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
14 | 13, 7 | grpo2inv 28300 | . . 3 ⊢ ((( +𝑣 ‘𝑈) ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴)) = 𝐴) |
15 | 12, 14 | sylan 582 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴)) = 𝐴) |
16 | 9, 11, 15 | 3eqtrd 2858 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 1c1 10530 -cneg 10863 GrpOpcgr 28258 invcgn 28260 NrmCVeccnv 28353 +𝑣 cpv 28354 BaseSetcba 28355 ·𝑠OLD cns 28356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-sub 10864 df-neg 10865 df-grpo 28262 df-gid 28263 df-ginv 28264 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-nmcv 28369 |
This theorem is referenced by: nvdif 28435 |
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