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Theorem grplactval 17286
Description: The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (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
grplactval ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))
Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔   𝐵,𝑎
Allowed substitution hints:   𝐵(𝑔)   𝐹(𝑔,𝑎)

Proof of Theorem grplactval
StepHypRef Expression
1 grplact.1 . . . 4 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
2 grplact.2 . . . 4 𝑋 = (Base‘𝐺)
31, 2grplactfval 17285 . . 3 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
43fveq1d 6090 . 2 (𝐴𝑋 → ((𝐹𝐴)‘𝐵) = ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵))
5 oveq2 6535 . . 3 (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵))
6 eqid 2609 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
7 ovex 6555 . . 3 (𝐴 + 𝐵) ∈ V
85, 6, 7fvmpt 6176 . 2 (𝐵𝑋 → ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵))
94, 8sylan9eq 2663 1 ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cmpt 4637  cfv 5790  (class class class)co 6527  Basecbs 15641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530
This theorem is referenced by:  cayleylem2  17602  dchrsum2  24710  sumdchr2  24712
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