Theorem ga0 18549

Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Assertion

Ref	Expression
ga0	⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Proof of Theorem ga0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5176	. . 3 ⊢ ∅ ∈ V
2	1	jctr 528	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3		f0 6560	. . . 4 ⊢ ∅:∅⟶∅
4		xp0 5991	. . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅
5	4	feq2i 6497	. . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
6	3, 5	mpbir 234	. . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅
7		ral0 4400	. . 3 ⊢ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))
8	6, 7	pm3.2i 474	. 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))))
9		eqid 2739	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2739	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		eqid 2739	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
12	9, 10, 11	isga 18542	. 2 ⊢ (∅ ∈ (𝐺 GrpAct ∅) ↔ ((𝐺 ∈ Grp ∧ ∅ ∈ V) ∧ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))))))
13	2, 8, 12	sylanblrc 593	1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∀wral 3054  Vcvv 3399  ∅c0 4212   × cxp 5524  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173  Basecbs 16589  +_gcplusg 16671  0_gc0g 16819  Grpcgrp 18222   GrpAct cga 18540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-map 8442  df-ga 18541

This theorem is referenced by: (None)