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Theorem grplactfval 17638
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 6772 . . 3 (𝑔 = 𝐴 → (𝑔 + 𝑎) = (𝐴 + 𝑎))
21mpteq2dv 4853 . 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 6314 . . . 4 (Base‘𝐺) ∈ V
64, 5eqeltri 2799 . . 3 𝑋 ∈ V
76mptex 6602 . 2 (𝑎𝑋 ↦ (𝐴 + 𝑎)) ∈ V
82, 3, 7fvmpt 6396 1 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  Vcvv 3304  cmpt 4837  cfv 6001  (class class class)co 6765  Basecbs 15980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768
This theorem is referenced by:  grplactval  17639  grplactcnv  17640  eqglact  17767  eqgen  17769  tgplacthmeo  22029  tgpconncompeqg  22037
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