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Theorem grplactval 18176
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 18175 . . 3 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
43fveq1d 6644 . 2 (𝐴𝑋 → ((𝐹𝐴)‘𝐵) = ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵))
5 oveq2 7137 . . 3 (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵))
6 eqid 2820 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
7 ovex 7162 . . 3 (𝐴 + 𝐵) ∈ V
85, 6, 7fvmpt 6740 . 2 (𝐵𝑋 → ((𝑎𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵))
94, 8sylan9eq 2875 1 ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cmpt 5118  cfv 6327  (class class class)co 7129  Basecbs 16458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7132
This theorem is referenced by:  cayleylem2  18516  dchrsum2  25827  sumdchr2  25829
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