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Mirrors > Home > MPE Home > Th. List > clmvz | Structured version Visualization version GIF version |
Description: Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
clmvz.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvz.m | ⊢ − = (-g‘𝑊) |
clmvz.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmvz.0 | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
clmvz | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
2 | clmgrp 23666 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp) | |
3 | clmvz.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | clmvz.0 | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | 3, 4 | grpidcl 18125 | . . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑉) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 0 ∈ 𝑉) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 0 ∈ 𝑉) |
8 | simpr 487 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
9 | eqid 2821 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | clmvz.m | . . . 4 ⊢ − = (-g‘𝑊) | |
11 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | clmvz.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | 3, 9, 10, 11, 12 | clmvsubval2 23708 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 0 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = ((-1 · 𝐴)(+g‘𝑊) 0 )) |
14 | 1, 7, 8, 13 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = ((-1 · 𝐴)(+g‘𝑊) 0 )) |
15 | eqid 2821 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
16 | 11, 15 | clmneg1 23680 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
18 | 3, 11, 12, 15 | clmvscl 23686 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉) |
19 | 1, 17, 8, 18 | syl3anc 1367 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉) |
20 | 3, 9, 4 | grprid 18128 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (-1 · 𝐴) ∈ 𝑉) → ((-1 · 𝐴)(+g‘𝑊) 0 ) = (-1 · 𝐴)) |
21 | 2, 19, 20 | syl2an2r 683 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴)(+g‘𝑊) 0 ) = (-1 · 𝐴)) |
22 | 14, 21 | eqtrd 2856 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 1c1 10532 -cneg 10865 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Grpcgrp 18097 -gcsg 18099 ℂModcclm 23660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-seq 13364 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-cnfld 20540 df-clm 23661 |
This theorem is referenced by: (None) |
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