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Theorem vcm 27271
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 27265 . . . 4 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
32adantr 481 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → 𝐺 ∈ GrpOp)
4 neg1cn 11069 . . . 4 -1 ∈ ℂ
5 vcm.2 . . . . 5 𝑆 = (2nd𝑊)
6 vcm.3 . . . . 5 𝑋 = ran 𝐺
71, 5, 6vccl 27258 . . . 4 ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
84, 7mp3an2 1409 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9 eqid 2626 . . . 4 (GId‘𝐺) = (GId‘𝐺)
106, 9grporid 27211 . . 3 ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
113, 8, 10syl2anc 692 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
12 simpr 477 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → 𝐴𝑋)
13 vcm.4 . . . . . . . 8 𝑀 = (inv‘𝐺)
146, 13grpoinvcl 27218 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
152, 14sylan 488 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
166grpoass 27197 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((-1𝑆𝐴) ∈ 𝑋𝐴𝑋 ∧ (𝑀𝐴) ∈ 𝑋)) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
173, 8, 12, 15, 16syl13anc 1325 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
181, 5, 6vcidOLD 27259 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
1918oveq2d 6621 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20 ax-1cn 9939 . . . . . . . . . 10 1 ∈ ℂ
21 1pneg1e0 11074 . . . . . . . . . 10 (1 + -1) = 0
2220, 4, 21addcomli 10173 . . . . . . . . 9 (-1 + 1) = 0
2322oveq1i 6615 . . . . . . . 8 ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
241, 5, 6vcdir 27261 . . . . . . . . . 10 ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
254, 24mp3anr1 1418 . . . . . . . . 9 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
2620, 25mpanr1 718 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
271, 5, 6, 9vc0 27269 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (0𝑆𝐴) = (GId‘𝐺))
2823, 26, 273eqtr3a 2684 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
2919, 28eqtr3d 2662 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
3029oveq1d 6620 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((GId‘𝐺)𝐺(𝑀𝐴)))
3117, 30eqtr3d 2662 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((GId‘𝐺)𝐺(𝑀𝐴)))
326, 9, 13grporinv 27221 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
332, 32sylan 488 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
3433oveq2d 6621 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
3531, 34eqtr3d 2662 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
366, 9grpolid 27210 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑀𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
373, 15, 36syl2anc 692 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
3835, 37eqtr3d 2662 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀𝐴))
3911, 38eqtr3d 2662 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  ran crn 5080  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  cc 9879  0cc0 9881  1c1 9882   + caddc 9884  -cneg 10212  GrpOpcgr 27183  GIdcgi 27184  invcgn 27185  CVecOLDcvc 27253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-ltxr 10024  df-sub 10213  df-neg 10214  df-grpo 27187  df-gid 27188  df-ginv 27189  df-ablo 27239  df-vc 27254
This theorem is referenced by:  nvinv  27334
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