Theorem gaf 19364

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.

Step	Hyp	Ref	Expression
1		gaf.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2769	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2769	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 19360	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simprbi 502	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
6	5	simpld 499	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   × cxp 5660  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  +_gcplusg 17309  0_gc0g 17491  Grpcgrp 18999   GrpAct cga 19358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8825  df-ga 19359

This theorem is referenced by:  gafo  19365  gass  19370  gasubg  19371  gacan  19374  gapm  19375  gastacos  19379  orbsta  19382  galactghm  19473  sylow2alem2  19687  fxpgaval  33427  mplvrpmmhm  33880  mplvrpmrhm  33881