Theorem ga0 19206

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 5257	. . 3 ⊢ ∅ ∈ V
2	1	jctr 524	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3		f0 6723	. . . 4 ⊢ ∅:∅⟶∅
4		xp0 6119	. . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅
5	4	feq2i 6662	. . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
6	3, 5	mpbir 231	. . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅
7		ral0 4472	. . 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 470	. 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))))
9		eqid 2729	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2729	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		eqid 2729	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
12	9, 10, 11	isga 19199	. 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 590	1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444  ∅c0 4292   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Grpcgrp 18841   GrpAct cga 19197

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-ga 19198

This theorem is referenced by: (None)