Theorem grplactval 19060

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 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
grplact.2	⊢ 𝑋 = (Base‘𝐺)

Assertion

Ref	Expression
grplactval	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘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

Step	Hyp	Ref	Expression
1		grplact.1	. . . 4 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
2		grplact.2	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
3	1, 2	grplactfval 19059	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
4	3	fveq1d 6908	. 2 ⊢ (𝐴 ∈ 𝑋 → ((𝐹‘𝐴)‘𝐵) = ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵))
5		oveq2 7439	. . 3 ⊢ (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵))
6		eqid 2737	. . 3 ⊢ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))
7		ovex 7464	. . 3 ⊢ (𝐴 + 𝐵) ∈ V
8	5, 6, 7	fvmpt 7016	. 2 ⊢ (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵))
9	4, 8	sylan9eq 2797	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  Basecbs 17247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434

This theorem is referenced by:  cayleylem2  19431  dchrsum2  27312  sumdchr2  27314