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

Theorem grplactfval 17717
Description: The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.)
Hypotheses
Ref Expression
grplact.1 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
grplact.2 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
grplactfval (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔
Allowed substitution hints:   𝐹(𝑔,𝑎)

Proof of Theorem grplactfval
StepHypRef Expression
1 oveq1 6820 . . 3 (𝑔 = 𝐴 → (𝑔 + 𝑎) = (𝐴 + 𝑎))
21mpteq2dv 4897 . 2 (𝑔 = 𝐴 → (𝑎𝑋 ↦ (𝑔 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3 grplact.1 . 2 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
4 grplact.2 . . . 4 𝑋 = (Base‘𝐺)
5 fvex 6362 . . . 4 (Base‘𝐺) ∈ V
64, 5eqeltri 2835 . . 3 𝑋 ∈ V
76mptex 6650 . 2 (𝑎𝑋 ↦ (𝐴 + 𝑎)) ∈ V
82, 3, 7fvmpt 6444 1 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  cfv 6049  (class class class)co 6813  Basecbs 16059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816
This theorem is referenced by:  grplactval  17718  grplactcnv  17719  eqglact  17846  eqgen  17848  tgplacthmeo  22108  tgpconncompeqg  22116
  Copyright terms: Public domain W3C validator