Theorem ga0 19473

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 5260	. . 3 ⊢ ∅ ∈ V
2	1	jctr 534	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3		f0 6751	. . . 4 ⊢ ∅:∅⟶∅
4		xp0 5747	. . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅
5	4	feq2i 6689	. . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
6	3, 5	mpbir 234	. . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅
7		ral0 4453	. . 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 476	. 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))))
9		eqid 2760	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2760	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		eqid 2760	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
12	9, 10, 11	isga 19466	. 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 602	1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4278   × cxp 5645  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  +_gcplusg 17389  0_gc0g 17571  Grpcgrp 19105   GrpAct cga 19464

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8827  df-ga 19465

This theorem is used by: (None)