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Theorem vcm 28286
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 28280 . . . 4 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
32adantr 481 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → 𝐺 ∈ GrpOp)
4 neg1cn 11745 . . . 4 -1 ∈ ℂ
5 vcm.2 . . . . 5 𝑆 = (2nd𝑊)
6 vcm.3 . . . . 5 𝑋 = ran 𝐺
71, 5, 6vccl 28273 . . . 4 ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
84, 7mp3an2 1442 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9 eqid 2826 . . . 4 (GId‘𝐺) = (GId‘𝐺)
106, 9grporid 28227 . . 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 28234 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
152, 14sylan 580 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝑀𝐴) ∈ 𝑋)
166grpoass 28213 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((-1𝑆𝐴) ∈ 𝑋𝐴𝑋 ∧ (𝑀𝐴) ∈ 𝑋)) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
173, 8, 12, 15, 16syl13anc 1366 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))))
181, 5, 6vcidOLD 28274 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
1918oveq2d 7166 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20 ax-1cn 10589 . . . . . . . . . 10 1 ∈ ℂ
21 1pneg1e0 11750 . . . . . . . . . 10 (1 + -1) = 0
2220, 4, 21addcomli 10826 . . . . . . . . 9 (-1 + 1) = 0
2322oveq1i 7160 . . . . . . . 8 ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
241, 5, 6vcdir 28276 . . . . . . . . . 10 ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
254, 24mp3anr1 1451 . . . . . . . . 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 28284 . . . . . . . 8 ((𝑊 ∈ CVecOLD𝐴𝑋) → (0𝑆𝐴) = (GId‘𝐺))
2823, 26, 273eqtr3a 2885 . . . . . . 7 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
2919, 28eqtr3d 2863 . . . . . 6 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
3029oveq1d 7165 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀𝐴)) = ((GId‘𝐺)𝐺(𝑀𝐴)))
3117, 30eqtr3d 2863 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((GId‘𝐺)𝐺(𝑀𝐴)))
326, 9, 13grporinv 28237 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
332, 32sylan 580 . . . . 5 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺(𝑀𝐴)) = (GId‘𝐺))
3433oveq2d 7166 . . . 4 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
3531, 34eqtr3d 2863 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
366, 9grpolid 28226 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑀𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
373, 15, 36syl2anc 584 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((GId‘𝐺)𝐺(𝑀𝐴)) = (𝑀𝐴))
3835, 37eqtr3d 2863 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀𝐴))
3911, 38eqtr3d 2863 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  ran crn 5555  cfv 6354  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  cc 10529  0cc0 10531  1c1 10532   + caddc 10534  -cneg 10865  GrpOpcgr 28199  GIdcgi 28200  invcgn 28201  CVecOLDcvc 28268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  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
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-ltxr 10674  df-sub 10866  df-neg 10867  df-grpo 28203  df-gid 28204  df-ginv 28205  df-ablo 28255  df-vc 28269
This theorem is referenced by:  nvinv  28349
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