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Theorem vcm 28280
Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vcm.1 𝐺 = (1st𝑊)
vcm.2 𝑆 = (2nd𝑊)
vcm.3 𝑋 = ran 𝐺
vcm.4 𝑀 = (inv‘𝐺)
Assertion
Ref Expression
vcm ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))

Proof of Theorem vcm
StepHypRef Expression
1 vcm.1 . . . . 5 𝐺 = (1st𝑊)
21vcgrp 28274 . . . 4 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
32adantr 481 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → 𝐺 ∈ GrpOp)
4 neg1cn 11739 . . . 4 -1 ∈ ℂ
5 vcm.2 . . . . 5 𝑆 = (2nd𝑊)
6 vcm.3 . . . . 5 𝑋 = ran 𝐺
71, 5, 6vccl 28267 . . . 4 ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
84, 7mp3an2 1440 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9 eqid 2818 . . . 4 (GId‘𝐺) = (GId‘𝐺)
106, 9grporid 28221 . . 3 ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
113, 8, 10syl2anc 584 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
12 simpr 485 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → 𝐴𝑋)
13 vcm.4 . . . . . . . 8 𝑀 = (inv‘𝐺)
146, 13grpoinvcl 28228 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
152, 14sylan 580 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
166grpoass 28207 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((-1𝑆𝐴) ∈ 𝑋𝐴𝑋 ∧ (𝑀𝐴) ∈ 𝑋)) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
173, 8, 12, 15, 16syl13anc 1364 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
181, 5, 6vcidOLD 28268 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
1918oveq2d 7161 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20 ax-1cn 10583 . . . . . . . . . 10 1 ∈ ℂ
21 1pneg1e0 11744 . . . . . . . . . 10 (1 + -1) = 0
2220, 4, 21addcomli 10820 . . . . . . . . 9 (-1 + 1) = 0
2322oveq1i 7155 . . . . . . . 8 ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
241, 5, 6vcdir 28270 . . . . . . . . . 10 ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
254, 24mp3anr1 1449 . . . . . . . . 9 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
2620, 25mpanr1 699 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
271, 5, 6, 9vc0 28278 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (0𝑆𝐴) = (GId‘𝐺))
2823, 26, 273eqtr3a 2877 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
2919, 28eqtr3d 2855 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
3029oveq1d 7160 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((GId‘𝐺)𝐺(𝑀𝐴)))
3117, 30eqtr3d 2855 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((GId‘𝐺)𝐺(𝑀𝐴)))
326, 9, 13grporinv 28231 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
332, 32sylan 580 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
3433oveq2d 7161 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
3531, 34eqtr3d 2855 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
366, 9grpolid 28220 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑀𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
373, 15, 36syl2anc 584 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
3835, 37eqtr3d 2855 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀𝐴))
3911, 38eqtr3d 2855 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  cc 10523  0cc0 10525  1c1 10526   + caddc 10528  -cneg 10859  GrpOpcgr 28193  GIdcgi 28194  invcgn 28195  CVecOLDcvc 28262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-ltxr 10668  df-sub 10860  df-neg 10861  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-vc 28263
This theorem is referenced by:  nvinv  28343
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