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Theorem grplactval 18970
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 18969 . . 3 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
43fveq1d 6887 . 2 (𝐴𝑋 → ((𝐹𝐴)‘𝐵) = ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵))
5 oveq2 7413 . . 3 (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵))
6 eqid 2726 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
7 ovex 7438 . . 3 (𝐴 + 𝐵) ∈ V
85, 6, 7fvmpt 6992 . 2 (𝐵𝑋 → ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵))
94, 8sylan9eq 2786 1 ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cmpt 5224  cfv 6537  (class class class)co 7405  Basecbs 17153
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408
This theorem is referenced by:  cayleylem2  19333  dchrsum2  27156  sumdchr2  27158
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