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Mirrors > Home > MPE Home > Th. List > clmnegneg | Structured version Visualization version GIF version |
Description: Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.) |
Ref | Expression |
---|---|
clmpm1dir.v | ⊢ 𝑉 = (Base‘𝑊) |
clmpm1dir.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmpm1dir.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
clmnegneg | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1mulneg1e1 11433 | . . 3 ⊢ (-1 · -1) = 1 | |
2 | 1 | oveq1i 6819 | . 2 ⊢ ((-1 · -1) · 𝐴) = (1 · 𝐴) |
3 | simpl 474 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
4 | eqid 2756 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2756 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 4, 5 | clmneg1 23078 | . . . 4 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
7 | 6 | adantr 472 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
8 | simpr 479 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
9 | clmpm1dir.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
10 | clmpm1dir.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | 9, 4, 10, 5 | clmvsass 23085 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (-1 ∈ (Base‘(Scalar‘𝑊)) ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉)) → ((-1 · -1) · 𝐴) = (-1 · (-1 · 𝐴))) |
12 | 3, 7, 7, 8, 11 | syl13anc 1479 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · -1) · 𝐴) = (-1 · (-1 · 𝐴))) |
13 | 9, 10 | clmvs1 23089 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (1 · 𝐴) = 𝐴) |
14 | 2, 12, 13 | 3eqtr3a 2814 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1628 ∈ wcel 2135 ‘cfv 6045 (class class class)co 6809 1c1 10125 · cmul 10129 -cneg 10455 Basecbs 16055 +gcplusg 16139 Scalarcsca 16142 ·𝑠 cvsca 16143 ℂModcclm 23058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-inf2 8707 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 ax-addf 10203 ax-mulf 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-int 4624 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-om 7227 df-1st 7329 df-2nd 7330 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-1o 7725 df-oadd 7729 df-er 7907 df-en 8118 df-dom 8119 df-sdom 8120 df-fin 8121 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-nn 11209 df-2 11267 df-3 11268 df-4 11269 df-5 11270 df-6 11271 df-7 11272 df-8 11273 df-9 11274 df-n0 11481 df-z 11566 df-dec 11682 df-uz 11876 df-fz 12516 df-seq 12992 df-struct 16057 df-ndx 16058 df-slot 16059 df-base 16061 df-sets 16062 df-ress 16063 df-plusg 16152 df-mulr 16153 df-starv 16154 df-tset 16158 df-ple 16159 df-ds 16162 df-unif 16163 df-0g 16300 df-mgm 17439 df-sgrp 17481 df-mnd 17492 df-grp 17622 df-minusg 17623 df-mulg 17738 df-subg 17788 df-cmn 18391 df-mgp 18686 df-ur 18698 df-ring 18745 df-cring 18746 df-subrg 18976 df-lmod 19063 df-cnfld 19945 df-clm 23059 |
This theorem is referenced by: clmnegsubdi2 23101 |
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