MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vcm Structured version   Visualization version   GIF version

Theorem vcm 29560
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 = (1st β€˜π‘Š)
vcm.2 𝑆 = (2nd β€˜π‘Š)
vcm.3 𝑋 = ran 𝐺
vcm.4 𝑀 = (invβ€˜πΊ)
Assertion
Ref Expression
vcm ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (-1𝑆𝐴) = (π‘€β€˜π΄))

Proof of Theorem vcm
StepHypRef Expression
1 vcm.1 . . . . 5 𝐺 = (1st β€˜π‘Š)
21vcgrp 29554 . . . 4 (π‘Š ∈ CVecOLD β†’ 𝐺 ∈ GrpOp)
32adantr 482 . . 3 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ GrpOp)
4 neg1cn 12272 . . . 4 -1 ∈ β„‚
5 vcm.2 . . . . 5 𝑆 = (2nd β€˜π‘Š)
6 vcm.3 . . . . 5 𝑋 = ran 𝐺
71, 5, 6vccl 29547 . . . 4 ((π‘Š ∈ CVecOLD ∧ -1 ∈ β„‚ ∧ 𝐴 ∈ 𝑋) β†’ (-1𝑆𝐴) ∈ 𝑋)
84, 7mp3an2 1450 . . 3 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (-1𝑆𝐴) ∈ 𝑋)
9 eqid 2733 . . . 4 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
106, 9grporid 29501 . . 3 ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(GIdβ€˜πΊ)) = (-1𝑆𝐴))
113, 8, 10syl2anc 585 . 2 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(GIdβ€˜πΊ)) = (-1𝑆𝐴))
12 simpr 486 . . . . . 6 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
13 vcm.4 . . . . . . . 8 𝑀 = (invβ€˜πΊ)
146, 13grpoinvcl 29508 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘€β€˜π΄) ∈ 𝑋)
152, 14sylan 581 . . . . . 6 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (π‘€β€˜π΄) ∈ 𝑋)
166grpoass 29487 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((-1𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘€β€˜π΄) ∈ 𝑋)) β†’ (((-1𝑆𝐴)𝐺𝐴)𝐺(π‘€β€˜π΄)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(π‘€β€˜π΄))))
173, 8, 12, 15, 16syl13anc 1373 . . . . 5 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (((-1𝑆𝐴)𝐺𝐴)𝐺(π‘€β€˜π΄)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(π‘€β€˜π΄))))
181, 5, 6vcidOLD 29548 . . . . . . . 8 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (1𝑆𝐴) = 𝐴)
1918oveq2d 7374 . . . . . . 7 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20 ax-1cn 11114 . . . . . . . . . 10 1 ∈ β„‚
21 1pneg1e0 12277 . . . . . . . . . 10 (1 + -1) = 0
2220, 4, 21addcomli 11352 . . . . . . . . 9 (-1 + 1) = 0
2322oveq1i 7368 . . . . . . . 8 ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
241, 5, 6vcdir 29550 . . . . . . . . . 10 ((π‘Š ∈ CVecOLD ∧ (-1 ∈ β„‚ ∧ 1 ∈ β„‚ ∧ 𝐴 ∈ 𝑋)) β†’ ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
254, 24mp3anr1 1459 . . . . . . . . 9 ((π‘Š ∈ CVecOLD ∧ (1 ∈ β„‚ ∧ 𝐴 ∈ 𝑋)) β†’ ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
2620, 25mpanr1 702 . . . . . . . 8 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
271, 5, 6, 9vc0 29558 . . . . . . . 8 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (0𝑆𝐴) = (GIdβ€˜πΊ))
2823, 26, 273eqtr3a 2797 . . . . . . 7 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GIdβ€˜πΊ))
2919, 28eqtr3d 2775 . . . . . 6 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺𝐴) = (GIdβ€˜πΊ))
3029oveq1d 7373 . . . . 5 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (((-1𝑆𝐴)𝐺𝐴)𝐺(π‘€β€˜π΄)) = ((GIdβ€˜πΊ)𝐺(π‘€β€˜π΄)))
3117, 30eqtr3d 2775 . . . 4 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(𝐴𝐺(π‘€β€˜π΄))) = ((GIdβ€˜πΊ)𝐺(π‘€β€˜π΄)))
326, 9, 13grporinv 29511 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘€β€˜π΄)) = (GIdβ€˜πΊ))
332, 32sylan 581 . . . . 5 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘€β€˜π΄)) = (GIdβ€˜πΊ))
3433oveq2d 7374 . . . 4 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(𝐴𝐺(π‘€β€˜π΄))) = ((-1𝑆𝐴)𝐺(GIdβ€˜πΊ)))
3531, 34eqtr3d 2775 . . 3 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘€β€˜π΄)) = ((-1𝑆𝐴)𝐺(GIdβ€˜πΊ)))
366, 9grpolid 29500 . . . 4 ((𝐺 ∈ GrpOp ∧ (π‘€β€˜π΄) ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘€β€˜π΄)) = (π‘€β€˜π΄))
373, 15, 36syl2anc 585 . . 3 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘€β€˜π΄)) = (π‘€β€˜π΄))
3835, 37eqtr3d 2775 . 2 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ ((-1𝑆𝐴)𝐺(GIdβ€˜πΊ)) = (π‘€β€˜π΄))
3911, 38eqtr3d 2775 1 ((π‘Š ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) β†’ (-1𝑆𝐴) = (π‘€β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  β„‚cc 11054  0cc0 11056  1c1 11057   + caddc 11059  -cneg 11391  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475  CVecOLDcvc 29542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-ltxr 11199  df-sub 11392  df-neg 11393  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-vc 29543
This theorem is referenced by:  nvinv  29623
  Copyright terms: Public domain W3C validator