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Theorem gaf 17774
Description: The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaf.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
gaf ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)

Proof of Theorem gaf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaf.1 . . . 4 𝑋 = (Base‘𝐺)
2 eqid 2651 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2651 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3isga 17770 . . 3 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 479 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
65simpld 474 1 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469   GrpAct cga 17768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ga 17769
This theorem is referenced by:  gafo  17775  gass  17780  gasubg  17781  gacan  17784  gapm  17785  gastacos  17789  orbsta  17792  galactghm  17869  sylow2alem2  18079
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