Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clmvsrinv | Structured version Visualization version GIF version |
Description: A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.) |
Ref | Expression |
---|---|
clmpm1dir.v | ⊢ 𝑉 = (Base‘𝑊) |
clmpm1dir.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmpm1dir.a | ⊢ + = (+g‘𝑊) |
clmvsrinv.0 | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
clmvsrinv | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmpm1dir.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2819 | . . . 4 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
3 | eqid 2819 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | clmpm1dir.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | 1, 2, 3, 4 | clmvneg1 23695 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) = ((invg‘𝑊)‘𝐴)) |
6 | 5 | oveq2d 7164 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = (𝐴 + ((invg‘𝑊)‘𝐴))) |
7 | clmgrp 23664 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp) | |
8 | clmpm1dir.a | . . . 4 ⊢ + = (+g‘𝑊) | |
9 | clmvsrinv.0 | . . . 4 ⊢ 0 = (0g‘𝑊) | |
10 | 1, 8, 9, 2 | grprinv 18145 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉) → (𝐴 + ((invg‘𝑊)‘𝐴)) = 0 ) |
11 | 7, 10 | sylan 582 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + ((invg‘𝑊)‘𝐴)) = 0 ) |
12 | 6, 11 | eqtrd 2854 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 1c1 10530 -cneg 10863 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 ℂModcclm 23658 |
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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 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 ax-pre-mulgt0 10606 ax-addf 10608 ax-mulf 10609 |
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-rmo 3144 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-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-seq 13362 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-cmn 18900 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-subrg 19525 df-lmod 19628 df-cnfld 20538 df-clm 23659 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |