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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 12308 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | nvnegneg.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | nvnegneg.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | 2, 3 | nvscl 29742 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋) |
5 | 1, 4 | mp3an2 1449 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋) |
6 | eqid 2731 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
7 | eqid 2731 | . . . 4 ⊢ (inv‘( +𝑣 ‘𝑈)) = (inv‘( +𝑣 ‘𝑈)) | |
8 | 2, 6, 3, 7 | nvinv 29755 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1𝑆𝐴) ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴))) |
9 | 5, 8 | syldan 591 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴))) |
10 | 2, 6, 3, 7 | nvinv 29755 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = ((inv‘( +𝑣 ‘𝑈))‘𝐴)) |
11 | 10 | fveq2d 6882 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘(-1𝑆𝐴)) = ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴))) |
12 | 6 | nvgrp 29733 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
13 | 2, 6 | bafval 29720 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
14 | 13, 7 | grpo2inv 29647 | . . 3 ⊢ ((( +𝑣 ‘𝑈) ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴)) = 𝐴) |
15 | 12, 14 | sylan 580 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((inv‘( +𝑣 ‘𝑈))‘((inv‘( +𝑣 ‘𝑈))‘𝐴)) = 𝐴) |
16 | 9, 11, 15 | 3eqtrd 2775 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 1c1 11093 -cneg 11427 GrpOpcgr 29605 invcgn 29607 NrmCVeccnv 29700 +𝑣 cpv 29701 BaseSetcba 29702 ·𝑠OLD cns 29703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-sub 11428 df-neg 11429 df-grpo 29609 df-gid 29610 df-ginv 29611 df-ablo 29661 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-nmcv 29716 |
This theorem is referenced by: nvdif 29782 |
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